:: MATRIXJ2 semantic presentation
begin
Lm1: for x, y being set
for f being Function st f is one-to-one & x in dom f holds
rng (f +* (x,y)) = ((rng f) \ {(f . x)}) \/ {y}
proof
let x, y be set ; ::_thesis: for f being Function st f is one-to-one & x in dom f holds
rng (f +* (x,y)) = ((rng f) \ {(f . x)}) \/ {y}
let f be Function; ::_thesis: ( f is one-to-one & x in dom f implies rng (f +* (x,y)) = ((rng f) \ {(f . x)}) \/ {y} )
assume that
A1: f is one-to-one and
A2: x in dom f ; ::_thesis: rng (f +* (x,y)) = ((rng f) \ {(f . x)}) \/ {y}
set xy = x .--> y;
( dom (x .--> y) = {x} & rng (x .--> y) = {y} ) by FUNCOP_1:8, FUNCOP_1:13;
then rng (f +* (x .--> y)) = (f .: ((dom f) \ {x})) \/ {y} by FRECHET:12
.= ((f .: (dom f)) \ (f .: {x})) \/ {y} by A1, FUNCT_1:64
.= ((rng f) \ (Im (f,x))) \/ {y} by RELAT_1:113
.= ((rng f) \ {(f . x)}) \/ {y} by A2, FUNCT_1:59 ;
hence rng (f +* (x,y)) = ((rng f) \ {(f . x)}) \/ {y} by A2, FUNCT_7:def_3; ::_thesis: verum
end;
definition
let K be Field;
let L be Element of K;
let n be Nat;
func Jordan_block (L,n) -> Matrix of K means :Def1: :: MATRIXJ2:def 1
( len it = n & width it = n & ( for i, j being Nat st [i,j] in Indices it holds
( ( i = j implies it * (i,j) = L ) & ( i + 1 = j implies it * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies it * (i,j) = 0. K ) ) ) );
existence
ex b1 being Matrix of K st
( len b1 = n & width b1 = n & ( for i, j being Nat st [i,j] in Indices b1 holds
( ( i = j implies b1 * (i,j) = L ) & ( i + 1 = j implies b1 * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies b1 * (i,j) = 0. K ) ) ) )
proof
defpred S1[ Nat, Nat, set ] means ( ( $1 = $2 implies $3 = L ) & ( $1 + 1 = $2 implies $3 = 1_ K ) & ( $1 <> $2 & $1 + 1 <> $2 implies $3 = 0. K ) );
reconsider N = n as Element of NAT by ORDINAL1:def_12;
A1: for i, j being Nat st [i,j] in [:(Seg N),(Seg N):] holds
ex x being Element of K st S1[i,j,x]
proof
let i, j be Nat; ::_thesis: ( [i,j] in [:(Seg N),(Seg N):] implies ex x being Element of K st S1[i,j,x] )
assume [i,j] in [:(Seg N),(Seg N):] ; ::_thesis: ex x being Element of K st S1[i,j,x]
percases ( i = j or i + 1 = j or ( i <> j & i + 1 <> j ) ) ;
supposeA2: i = j ; ::_thesis: ex x being Element of K st S1[i,j,x]
take L ; ::_thesis: S1[i,j,L]
thus S1[i,j,L] by A2; ::_thesis: verum
end;
supposeA3: i + 1 = j ; ::_thesis: ex x being Element of K st S1[i,j,x]
take 1_ K ; ::_thesis: S1[i,j, 1_ K]
thus S1[i,j, 1_ K] by A3; ::_thesis: verum
end;
supposeA4: ( i <> j & i + 1 <> j ) ; ::_thesis: ex x being Element of K st S1[i,j,x]
take 0. K ; ::_thesis: S1[i,j, 0. K]
thus S1[i,j, 0. K] by A4; ::_thesis: verum
end;
end;
end;
consider M being Matrix of N,K such that
A5: for i, j being Nat st [i,j] in Indices M holds
S1[i,j,M * (i,j)] from MATRIX_1:sch_2(A1);
take M ; ::_thesis: ( len M = n & width M = n & ( for i, j being Nat st [i,j] in Indices M holds
( ( i = j implies M * (i,j) = L ) & ( i + 1 = j implies M * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies M * (i,j) = 0. K ) ) ) )
thus ( len M = n & width M = n & ( for i, j being Nat st [i,j] in Indices M holds
( ( i = j implies M * (i,j) = L ) & ( i + 1 = j implies M * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies M * (i,j) = 0. K ) ) ) ) by A5, MATRIX_1:24; ::_thesis: verum
end;
uniqueness
for b1, b2 being Matrix of K st len b1 = n & width b1 = n & ( for i, j being Nat st [i,j] in Indices b1 holds
( ( i = j implies b1 * (i,j) = L ) & ( i + 1 = j implies b1 * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies b1 * (i,j) = 0. K ) ) ) & len b2 = n & width b2 = n & ( for i, j being Nat st [i,j] in Indices b2 holds
( ( i = j implies b2 * (i,j) = L ) & ( i + 1 = j implies b2 * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies b2 * (i,j) = 0. K ) ) ) holds
b1 = b2
proof
let M1, M2 be Matrix of K; ::_thesis: ( len M1 = n & width M1 = n & ( for i, j being Nat st [i,j] in Indices M1 holds
( ( i = j implies M1 * (i,j) = L ) & ( i + 1 = j implies M1 * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies M1 * (i,j) = 0. K ) ) ) & len M2 = n & width M2 = n & ( for i, j being Nat st [i,j] in Indices M2 holds
( ( i = j implies M2 * (i,j) = L ) & ( i + 1 = j implies M2 * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies M2 * (i,j) = 0. K ) ) ) implies M1 = M2 )
assume that
A6: ( len M1 = n & width M1 = n ) and
A7: for i, j being Nat st [i,j] in Indices M1 holds
( ( i = j implies M1 * (i,j) = L ) & ( i + 1 = j implies M1 * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies M1 * (i,j) = 0. K ) ) and
A8: ( len M2 = n & width M2 = n ) and
A9: for i, j being Nat st [i,j] in Indices M2 holds
( ( i = j implies M2 * (i,j) = L ) & ( i + 1 = j implies M2 * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies M2 * (i,j) = 0. K ) ) ; ::_thesis: M1 = M2
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_M1_holds_
M1_*_(i,j)_=_M2_*_(i,j)
let i, j be Nat; ::_thesis: ( [i,j] in Indices M1 implies M1 * (i,j) = M2 * (i,j) )
assume A10: [i,j] in Indices M1 ; ::_thesis: M1 * (i,j) = M2 * (i,j)
A11: Indices M1 = [:(Seg n),(Seg n):] by A6, FINSEQ_1:def_3
.= Indices M2 by A8, FINSEQ_1:def_3 ;
( i = j or i + 1 = j or ( i <> j & i + 1 <> j ) ) ;
then ( ( M1 * (i,j) = L & M2 * (i,j) = L ) or ( M1 * (i,j) = 1_ K & M2 * (i,j) = 1_ K ) or ( M1 * (i,j) = 0. K & M2 * (i,j) = 0. K ) ) by A7, A9, A10, A11;
hence M1 * (i,j) = M2 * (i,j) ; ::_thesis: verum
end;
hence M1 = M2 by A6, A8, MATRIX_1:21; ::_thesis: verum
end;
end;
:: deftheorem Def1 defines Jordan_block MATRIXJ2:def_1_:_
for K being Field
for L being Element of K
for n being Nat
for b4 being Matrix of K holds
( b4 = Jordan_block (L,n) iff ( len b4 = n & width b4 = n & ( for i, j being Nat st [i,j] in Indices b4 holds
( ( i = j implies b4 * (i,j) = L ) & ( i + 1 = j implies b4 * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies b4 * (i,j) = 0. K ) ) ) ) );
definition
let K be Field;
let L be Element of K;
let n be Nat;
:: original: Jordan_block
redefine func Jordan_block (L,n) -> Upper_Triangular_Matrix of n,K;
coherence
Jordan_block (L,n) is Upper_Triangular_Matrix of n,K
proof
( len (Jordan_block (L,n)) = n & width (Jordan_block (L,n)) = n ) by Def1;
then reconsider LBn = Jordan_block (L,n) as Matrix of n,K by MATRIX_2:7;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_LBn_&_i_>_j_holds_
LBn_*_(i,j)_=_0._K
let i, j be Nat; ::_thesis: ( [i,j] in Indices LBn & i > j implies LBn * (i,j) = 0. K )
assume A1: [i,j] in Indices LBn ; ::_thesis: ( i > j implies LBn * (i,j) = 0. K )
assume A2: i > j ; ::_thesis: LBn * (i,j) = 0. K
then i + 1 > j by NAT_1:13;
hence LBn * (i,j) = 0. K by A1, A2, Def1; ::_thesis: verum
end;
hence Jordan_block (L,n) is Upper_Triangular_Matrix of n,K by MATRIX_2:def_3; ::_thesis: verum
end;
end;
theorem Th1: :: MATRIXJ2:1
for n being Nat
for K being Field
for L being Element of K holds diagonal_of_Matrix (Jordan_block (L,n)) = n |-> L
proof
let n be Nat; ::_thesis: for K being Field
for L being Element of K holds diagonal_of_Matrix (Jordan_block (L,n)) = n |-> L
let K be Field; ::_thesis: for L being Element of K holds diagonal_of_Matrix (Jordan_block (L,n)) = n |-> L
let L be Element of K; ::_thesis: diagonal_of_Matrix (Jordan_block (L,n)) = n |-> L
set B = Jordan_block (L,n);
A1: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_n_holds_
(diagonal_of_Matrix_(Jordan_block_(L,n)))_._i_=_(n_|->_L)_._i
A2: [:(Seg n),(Seg n):] = Indices (Jordan_block (L,n)) by MATRIX_1:24;
let i be Nat; ::_thesis: ( 1 <= i & i <= n implies (diagonal_of_Matrix (Jordan_block (L,n))) . i = (n |-> L) . i )
assume ( 1 <= i & i <= n ) ; ::_thesis: (diagonal_of_Matrix (Jordan_block (L,n))) . i = (n |-> L) . i
then A3: i in Seg n by FINSEQ_1:1;
then A4: [i,i] in [:(Seg n),(Seg n):] by ZFMISC_1:87;
thus (diagonal_of_Matrix (Jordan_block (L,n))) . i = (Jordan_block (L,n)) * (i,i) by A3, MATRIX_3:def_10
.= L by A4, A2, Def1
.= (n |-> L) . i by A3, FINSEQ_2:57 ; ::_thesis: verum
end;
( len (diagonal_of_Matrix (Jordan_block (L,n))) = n & len (n |-> L) = n ) by CARD_1:def_7, MATRIX_3:def_10;
hence diagonal_of_Matrix (Jordan_block (L,n)) = n |-> L by A1, FINSEQ_1:14; ::_thesis: verum
end;
theorem Th2: :: MATRIXJ2:2
for n being Nat
for K being Field
for L being Element of K holds Det (Jordan_block (L,n)) = (power K) . (L,n)
proof
let n be Nat; ::_thesis: for K being Field
for L being Element of K holds Det (Jordan_block (L,n)) = (power K) . (L,n)
let K be Field; ::_thesis: for L being Element of K holds Det (Jordan_block (L,n)) = (power K) . (L,n)
let L be Element of K; ::_thesis: Det (Jordan_block (L,n)) = (power K) . (L,n)
thus Det (Jordan_block (L,n)) = the multF of K $$ (diagonal_of_Matrix (Jordan_block (L,n))) by MATRIX13:7
.= Product (n |-> L) by Th1
.= (power K) . (L,n) by MATRIXJ1:5 ; ::_thesis: verum
end;
theorem Th3: :: MATRIXJ2:3
for n being Nat
for K being Field
for L being Element of K holds
( Jordan_block (L,n) is invertible iff ( n = 0 or L <> 0. K ) )
proof
let n be Nat; ::_thesis: for K being Field
for L being Element of K holds
( Jordan_block (L,n) is invertible iff ( n = 0 or L <> 0. K ) )
let K be Field; ::_thesis: for L being Element of K holds
( Jordan_block (L,n) is invertible iff ( n = 0 or L <> 0. K ) )
let L be Element of K; ::_thesis: ( Jordan_block (L,n) is invertible iff ( n = 0 or L <> 0. K ) )
set B = Jordan_block (L,n);
A1: ( not Jordan_block (L,n) is invertible or L <> 0. K or n = 0 )
proof
assume Jordan_block (L,n) is invertible ; ::_thesis: ( L <> 0. K or n = 0 )
then A2: Det (Jordan_block (L,n)) <> 0. K by LAPLACE:34;
assume A3: L = 0. K ; ::_thesis: n = 0
assume n <> 0 ; ::_thesis: contradiction
then A4: n in Seg n by FINSEQ_1:3;
then ( dom (n |-> L) = Seg n & (n |-> L) . n = L ) by FINSEQ_2:57, FINSEQ_2:124;
then 0. K = Product (n |-> L) by A3, A4, FVSUM_1:82
.= (power K) . (L,n) by MATRIXJ1:5 ;
hence contradiction by A2, Th2; ::_thesis: verum
end;
( ( n = 0 or L <> 0. K ) implies Jordan_block (L,n) is invertible )
proof
assume A5: ( n = 0 or L <> 0. K ) ; ::_thesis: Jordan_block (L,n) is invertible
assume not Jordan_block (L,n) is invertible ; ::_thesis: contradiction
then 0. K = Det (Jordan_block (L,n)) by LAPLACE:34
.= (power K) . (L,n) by Th2
.= Product (n |-> L) by MATRIXJ1:5 ;
then A6: ex k being Element of NAT st
( k in dom (n |-> L) & (n |-> L) . k = 0. K ) by FVSUM_1:82;
dom (n |-> L) = Seg n by FINSEQ_2:124;
hence contradiction by A5, A6, FINSEQ_2:57; ::_thesis: verum
end;
hence ( Jordan_block (L,n) is invertible iff ( n = 0 or L <> 0. K ) ) by A1; ::_thesis: verum
end;
theorem Th4: :: MATRIXJ2:4
for i, n being Nat
for K being Field
for L being Element of K st i in Seg n & i <> n holds
Line ((Jordan_block (L,n)),i) = (L * (Line ((1. (K,n)),i))) + (Line ((1. (K,n)),(i + 1)))
proof
let i, n be Nat; ::_thesis: for K being Field
for L being Element of K st i in Seg n & i <> n holds
Line ((Jordan_block (L,n)),i) = (L * (Line ((1. (K,n)),i))) + (Line ((1. (K,n)),(i + 1)))
let K be Field; ::_thesis: for L being Element of K st i in Seg n & i <> n holds
Line ((Jordan_block (L,n)),i) = (L * (Line ((1. (K,n)),i))) + (Line ((1. (K,n)),(i + 1)))
let L be Element of K; ::_thesis: ( i in Seg n & i <> n implies Line ((Jordan_block (L,n)),i) = (L * (Line ((1. (K,n)),i))) + (Line ((1. (K,n)),(i + 1))) )
assume that
A1: i in Seg n and
A2: i <> n ; ::_thesis: Line ((Jordan_block (L,n)),i) = (L * (Line ((1. (K,n)),i))) + (Line ((1. (K,n)),(i + 1)))
reconsider N = n as Element of NAT by ORDINAL1:def_12;
set J = Jordan_block (L,n);
set i1 = i + 1;
set ONE = 1. (K,n);
set Li = Line ((1. (K,n)),i);
set Li1 = Line ((1. (K,n)),(i + 1));
set LJ = Line ((Jordan_block (L,n)),i);
A3: width (1. (K,n)) = n by MATRIX_1:24;
A4: Indices (1. (K,n)) = Indices (Jordan_block (L,n)) by MATRIX_1:26;
reconsider Li = Line ((1. (K,n)),i), Li1 = Line ((1. (K,n)),(i + 1)), LJ = Line ((Jordan_block (L,n)),i) as Element of N -tuples_on the carrier of K by MATRIX_1:24;
A5: Indices (1. (K,n)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
A6: width (Jordan_block (L,n)) = n by MATRIX_1:24;
now__::_thesis:_for_j_being_Nat_st_j_in_Seg_n_holds_
((L_*_Li)_+_Li1)_._j_=_LJ_._j
let j be Nat; ::_thesis: ( j in Seg n implies ((L * Li) + Li1) . j = LJ . j )
assume A7: j in Seg n ; ::_thesis: ((L * Li) + Li1) . j = LJ . j
Li . j = (1. (K,n)) * (i,j) by A3, A7, MATRIX_1:def_7;
then A8: (L * Li) . j = L * ((1. (K,n)) * (i,j)) by A7, FVSUM_1:51;
A9: [i,j] in [:(Seg n),(Seg n):] by A1, A7, ZFMISC_1:87;
i <= n by A1, FINSEQ_1:1;
then i < n by A2, XXREAL_0:1;
then ( 1 <= i + 1 & i + 1 <= n ) by NAT_1:11, NAT_1:13;
then i + 1 in Seg n ;
then A10: [(i + 1),j] in [:(Seg n),(Seg n):] by A7, ZFMISC_1:87;
Li1 . j = (1. (K,n)) * ((i + 1),j) by A3, A7, MATRIX_1:def_7;
then A11: ((L * Li) + Li1) . j = (L * ((1. (K,n)) * (i,j))) + ((1. (K,n)) * ((i + 1),j)) by A7, A8, FVSUM_1:18;
A12: LJ . j = (Jordan_block (L,n)) * (i,j) by A6, A7, MATRIX_1:def_7;
now__::_thesis:_LJ_._j_=_((L_*_Li)_+_Li1)_._j
percases ( i = j or i + 1 = j or ( i <> j & i + 1 <> j ) ) ;
supposeA13: i = j ; ::_thesis: LJ . j = ((L * Li) + Li1) . j
then A14: i + 1 > j by NAT_1:13;
thus LJ . j = L by A5, A4, A9, A12, A13, Def1
.= L + (0. K) by RLVECT_1:def_4
.= (L * (1_ K)) + (0. K) by VECTSP_1:def_8
.= (L * ((1. (K,n)) * (i,j))) + (0. K) by A5, A9, A13, MATRIX_1:def_11
.= ((L * Li) + Li1) . j by A5, A10, A11, A14, MATRIX_1:def_11 ; ::_thesis: verum
end;
supposeA15: i + 1 = j ; ::_thesis: LJ . j = ((L * Li) + Li1) . j
then A16: i < j by NAT_1:13;
thus LJ . j = 1_ K by A5, A4, A9, A12, A15, Def1
.= (0. K) + (1_ K) by RLVECT_1:def_4
.= (L * (0. K)) + (1_ K) by VECTSP_1:7
.= (L * ((1. (K,n)) * (i,j))) + (1_ K) by A5, A9, A16, MATRIX_1:def_11
.= ((L * Li) + Li1) . j by A5, A10, A11, A15, MATRIX_1:def_11 ; ::_thesis: verum
end;
supposeA17: ( i <> j & i + 1 <> j ) ; ::_thesis: LJ . j = ((L * Li) + Li1) . j
hence LJ . j = 0. K by A5, A4, A9, A12, Def1
.= (0. K) + (0. K) by RLVECT_1:def_4
.= (L * (0. K)) + (0. K) by VECTSP_1:7
.= (L * ((1. (K,n)) * (i,j))) + (0. K) by A5, A9, A17, MATRIX_1:def_11
.= ((L * Li) + Li1) . j by A5, A10, A11, A17, MATRIX_1:def_11 ;
::_thesis: verum
end;
end;
end;
hence ((L * Li) + Li1) . j = LJ . j ; ::_thesis: verum
end;
hence Line ((Jordan_block (L,n)),i) = (L * (Line ((1. (K,n)),i))) + (Line ((1. (K,n)),(i + 1))) by FINSEQ_2:119; ::_thesis: verum
end;
theorem Th5: :: MATRIXJ2:5
for n being Nat
for K being Field
for L being Element of K holds Line ((Jordan_block (L,n)),n) = L * (Line ((1. (K,n)),n))
proof
let n be Nat; ::_thesis: for K being Field
for L being Element of K holds Line ((Jordan_block (L,n)),n) = L * (Line ((1. (K,n)),n))
let K be Field; ::_thesis: for L being Element of K holds Line ((Jordan_block (L,n)),n) = L * (Line ((1. (K,n)),n))
let L be Element of K; ::_thesis: Line ((Jordan_block (L,n)),n) = L * (Line ((1. (K,n)),n))
set ONE = 1. (K,n);
set Ln = Line ((1. (K,n)),n);
set J = Jordan_block (L,n);
set LJ = Line ((Jordan_block (L,n)),n);
reconsider N = n as Element of NAT by ORDINAL1:def_12;
A1: width (Jordan_block (L,n)) = n by MATRIX_1:24;
A2: Indices (1. (K,n)) = Indices (Jordan_block (L,n)) by MATRIX_1:26;
reconsider Ln = Line ((1. (K,n)),n), LJ = Line ((Jordan_block (L,n)),n) as Element of N -tuples_on the carrier of K by MATRIX_1:24;
A3: Indices (1. (K,n)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
A4: width (1. (K,n)) = n by MATRIX_1:24;
now__::_thesis:_for_j_being_Nat_st_j_in_Seg_n_holds_
(L_*_Ln)_._j_=_LJ_._j
let j be Nat; ::_thesis: ( j in Seg n implies (L * Ln) . j = LJ . j )
assume A5: j in Seg n ; ::_thesis: (L * Ln) . j = LJ . j
n <> 0 by A5;
then n in Seg n by FINSEQ_1:3;
then A6: [n,j] in [:(Seg n),(Seg n):] by A5, ZFMISC_1:87;
Ln . j = (1. (K,n)) * (n,j) by A4, A5, MATRIX_1:def_7;
then A7: (L * Ln) . j = L * ((1. (K,n)) * (n,j)) by A5, FVSUM_1:51;
A8: LJ . j = (Jordan_block (L,n)) * (n,j) by A1, A5, MATRIX_1:def_7;
now__::_thesis:_LJ_._j_=_(L_*_Ln)_._j
percases ( n = j or n + 1 = j or ( n <> j & n + 1 <> j ) ) ;
supposeA9: n = j ; ::_thesis: LJ . j = (L * Ln) . j
hence LJ . j = L by A3, A2, A6, A8, Def1
.= L * (1_ K) by VECTSP_1:def_8
.= (L * Ln) . j by A3, A6, A7, A9, MATRIX_1:def_11 ;
::_thesis: verum
end;
suppose n + 1 = j ; ::_thesis: (L * Ln) . j = LJ . j
then j > n by NAT_1:13;
hence (L * Ln) . j = LJ . j by A5, FINSEQ_1:1; ::_thesis: verum
end;
supposeA10: ( n <> j & n + 1 <> j ) ; ::_thesis: LJ . j = (L * Ln) . j
hence LJ . j = 0. K by A3, A2, A6, A8, Def1
.= L * (0. K) by VECTSP_1:7
.= (L * Ln) . j by A3, A6, A7, A10, MATRIX_1:def_11 ;
::_thesis: verum
end;
end;
end;
hence (L * Ln) . j = LJ . j ; ::_thesis: verum
end;
hence Line ((Jordan_block (L,n)),n) = L * (Line ((1. (K,n)),n)) by FINSEQ_2:119; ::_thesis: verum
end;
theorem Th6: :: MATRIXJ2:6
for n, i being Nat
for K being Field
for L being Element of K
for F being Element of n -tuples_on the carrier of K st i in Seg n holds
( ( i <> n implies (Line ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i + 1)) ) & ( i = n implies (Line ((Jordan_block (L,n)),i)) "*" F = L * (F /. i) ) )
proof
let n, i be Nat; ::_thesis: for K being Field
for L being Element of K
for F being Element of n -tuples_on the carrier of K st i in Seg n holds
( ( i <> n implies (Line ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i + 1)) ) & ( i = n implies (Line ((Jordan_block (L,n)),i)) "*" F = L * (F /. i) ) )
let K be Field; ::_thesis: for L being Element of K
for F being Element of n -tuples_on the carrier of K st i in Seg n holds
( ( i <> n implies (Line ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i + 1)) ) & ( i = n implies (Line ((Jordan_block (L,n)),i)) "*" F = L * (F /. i) ) )
let L be Element of K; ::_thesis: for F being Element of n -tuples_on the carrier of K st i in Seg n holds
( ( i <> n implies (Line ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i + 1)) ) & ( i = n implies (Line ((Jordan_block (L,n)),i)) "*" F = L * (F /. i) ) )
let F be Element of n -tuples_on the carrier of K; ::_thesis: ( i in Seg n implies ( ( i <> n implies (Line ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i + 1)) ) & ( i = n implies (Line ((Jordan_block (L,n)),i)) "*" F = L * (F /. i) ) ) )
assume A1: i in Seg n ; ::_thesis: ( ( i <> n implies (Line ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i + 1)) ) & ( i = n implies (Line ((Jordan_block (L,n)),i)) "*" F = L * (F /. i) ) )
set J = Jordan_block (L,n);
A2: width (Jordan_block (L,n)) = n by MATRIX_1:24;
then A3: (Line ((Jordan_block (L,n)),i)) . i = (Jordan_block (L,n)) * (i,i) by A1, MATRIX_1:def_7;
A4: Indices (Jordan_block (L,n)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
then A5: [i,i] in Indices (Jordan_block (L,n)) by A1, ZFMISC_1:87;
reconsider N = n as Element of NAT by ORDINAL1:def_12;
set Li = Line ((Jordan_block (L,n)),i);
reconsider Li = Line ((Jordan_block (L,n)),i), f = F as Element of N -tuples_on the carrier of K by MATRIX_1:24;
A6: dom f = Seg n by FINSEQ_2:124;
then A7: f . i = f /. i by A1, PARTFUN1:def_6;
A8: dom (mlt (Li,f)) = Seg n by FINSEQ_2:124;
thus ( i <> n implies (Line ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i + 1)) ) ::_thesis: ( i = n implies (Line ((Jordan_block (L,n)),i)) "*" F = L * (F /. i) )
proof
A9: now__::_thesis:_for_j_being_Nat_st_j_in_Seg_n_&_j_<>_i_&_j_<>_i_+_1_holds_
(mlt_(Li,f))_._j_=_0._K
let j be Nat; ::_thesis: ( j in Seg n & j <> i & j <> i + 1 implies (mlt (Li,f)) . j = 0. K )
assume that
A10: j in Seg n and
A11: ( j <> i & j <> i + 1 ) ; ::_thesis: (mlt (Li,f)) . j = 0. K
[i,j] in Indices (Jordan_block (L,n)) by A1, A4, A10, ZFMISC_1:87;
then A12: 0. K = (Jordan_block (L,n)) * (i,j) by A11, Def1
.= Li . j by A2, A10, MATRIX_1:def_7 ;
f . j = f /. j by A6, A10, PARTFUN1:def_6;
hence (mlt (Li,f)) . j = (0. K) * (f /. j) by A10, A12, FVSUM_1:61
.= 0. K by VECTSP_1:6 ;
::_thesis: verum
end;
A13: [i,i] in Indices (Jordan_block (L,n)) by A1, A4, ZFMISC_1:87;
assume A14: i <> n ; ::_thesis: (Line ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i + 1))
i <= n by A1, FINSEQ_1:1;
then i < n by A14, XXREAL_0:1;
then ( 1 <= i + 1 & i + 1 <= n ) by NAT_1:11, NAT_1:13;
then A15: i + 1 in Seg n ;
then A16: [i,(i + 1)] in Indices (Jordan_block (L,n)) by A1, A4, ZFMISC_1:87;
A17: ( f . i = f /. i & (Jordan_block (L,n)) * (i,i) = Li . i ) by A1, A2, A6, MATRIX_1:def_7, PARTFUN1:def_6;
A18: (mlt (Li,f)) /. i = (mlt (Li,f)) . i by A1, A8, PARTFUN1:def_6
.= ((Jordan_block (L,n)) * (i,i)) * (f /. i) by A1, A17, FVSUM_1:61
.= L * (f /. i) by A13, Def1 ;
A19: i + 1 > i by NAT_1:13;
A20: ( f . (i + 1) = f /. (i + 1) & (Jordan_block (L,n)) * (i,(i + 1)) = Li . (i + 1) ) by A2, A6, A15, MATRIX_1:def_7, PARTFUN1:def_6;
(mlt (Li,f)) /. (i + 1) = (mlt (Li,f)) . (i + 1) by A8, A15, PARTFUN1:def_6
.= ((Jordan_block (L,n)) * (i,(i + 1))) * (f /. (i + 1)) by A15, A20, FVSUM_1:61
.= (1_ K) * (f /. (i + 1)) by A16, Def1
.= f /. (i + 1) by VECTSP_1:def_4 ;
hence (Line ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i + 1)) by A1, A8, A15, A19, A18, A9, MATRIX15:7; ::_thesis: verum
end;
assume A21: i = n ; ::_thesis: (Line ((Jordan_block (L,n)),i)) "*" F = L * (F /. i)
now__::_thesis:_for_j_being_Nat_st_j_in_Seg_n_&_j_<>_i_holds_
(mlt_((Line_((Jordan_block_(L,n)),i)),f))_._j_=_0._K
let j be Nat; ::_thesis: ( j in Seg n & j <> i implies (mlt ((Line ((Jordan_block (L,n)),i)),f)) . j = 0. K )
assume that
A22: j in Seg n and
A23: j <> i ; ::_thesis: (mlt ((Line ((Jordan_block (L,n)),i)),f)) . j = 0. K
A24: f . j = f /. j by A6, A22, PARTFUN1:def_6;
j <= n by A22, FINSEQ_1:1;
then A25: j < i + 1 by A21, NAT_1:13;
A26: [i,j] in Indices (Jordan_block (L,n)) by A1, A4, A22, ZFMISC_1:87;
(Line ((Jordan_block (L,n)),i)) . j = (Jordan_block (L,n)) * (i,j) by A2, A22, MATRIX_1:def_7
.= 0. K by A23, A25, A26, Def1 ;
hence (mlt ((Line ((Jordan_block (L,n)),i)),f)) . j = (0. K) * (f /. j) by A8, A22, A24, FVSUM_1:61
.= 0. K by VECTSP_1:6 ;
::_thesis: verum
end;
hence (Line ((Jordan_block (L,n)),i)) "*" F = (mlt ((Line ((Jordan_block (L,n)),i)),f)) . i by A1, A8, MATRIX_3:12
.= ((Jordan_block (L,n)) * (i,i)) * (f /. i) by A1, A8, A3, A7, FVSUM_1:61
.= L * (F /. i) by A5, Def1 ;
::_thesis: verum
end;
theorem Th7: :: MATRIXJ2:7
for n, i being Nat
for K being Field
for L being Element of K
for F being Element of n -tuples_on the carrier of K st i in Seg n holds
( ( i = 1 implies (Col ((Jordan_block (L,n)),i)) "*" F = L * (F /. i) ) & ( i <> 1 implies (Col ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i - 1)) ) )
proof
let n, i be Nat; ::_thesis: for K being Field
for L being Element of K
for F being Element of n -tuples_on the carrier of K st i in Seg n holds
( ( i = 1 implies (Col ((Jordan_block (L,n)),i)) "*" F = L * (F /. i) ) & ( i <> 1 implies (Col ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i - 1)) ) )
let K be Field; ::_thesis: for L being Element of K
for F being Element of n -tuples_on the carrier of K st i in Seg n holds
( ( i = 1 implies (Col ((Jordan_block (L,n)),i)) "*" F = L * (F /. i) ) & ( i <> 1 implies (Col ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i - 1)) ) )
let L be Element of K; ::_thesis: for F being Element of n -tuples_on the carrier of K st i in Seg n holds
( ( i = 1 implies (Col ((Jordan_block (L,n)),i)) "*" F = L * (F /. i) ) & ( i <> 1 implies (Col ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i - 1)) ) )
set J = Jordan_block (L,n);
set Ci = Col ((Jordan_block (L,n)),i);
reconsider N = n as Element of NAT by ORDINAL1:def_12;
let F be Element of n -tuples_on the carrier of K; ::_thesis: ( i in Seg n implies ( ( i = 1 implies (Col ((Jordan_block (L,n)),i)) "*" F = L * (F /. i) ) & ( i <> 1 implies (Col ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i - 1)) ) ) )
assume A1: i in Seg n ; ::_thesis: ( ( i = 1 implies (Col ((Jordan_block (L,n)),i)) "*" F = L * (F /. i) ) & ( i <> 1 implies (Col ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i - 1)) ) )
A2: i >= 1 by A1, FINSEQ_1:1;
then reconsider i1 = i - 1 as Element of NAT by NAT_1:21;
A3: ( len (Jordan_block (L,n)) = n & dom (Jordan_block (L,n)) = Seg (len (Jordan_block (L,n))) ) by FINSEQ_1:def_3, MATRIX_1:24;
then A4: (Col ((Jordan_block (L,n)),i)) . i = (Jordan_block (L,n)) * (i,i) by A1, MATRIX_1:def_8;
A5: i1 + 1 >= i1 by NAT_1:11;
n >= i by A1, FINSEQ_1:1;
then A6: n >= i1 by A5, XXREAL_0:2;
A7: Indices (Jordan_block (L,n)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
then A8: [i,i] in Indices (Jordan_block (L,n)) by A1, ZFMISC_1:87;
reconsider Ci = Col ((Jordan_block (L,n)),i), f = F as Element of N -tuples_on the carrier of K by MATRIX_1:24;
A9: dom f = Seg n by FINSEQ_2:124;
then A10: f . i = f /. i by A1, PARTFUN1:def_6;
A11: dom (mlt (Ci,f)) = Seg n by FINSEQ_2:124;
then A12: (mlt (Ci,f)) /. i = (mlt (Ci,f)) . i by A1, PARTFUN1:def_6
.= ((Jordan_block (L,n)) * (i,i)) * (f /. i) by A1, A4, A10, FVSUM_1:61
.= L * (f /. i) by A8, Def1 ;
thus ( i = 1 implies (Col ((Jordan_block (L,n)),i)) "*" F = L * (F /. i) ) ::_thesis: ( i <> 1 implies (Col ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i - 1)) )
proof
A13: ( (Col ((Jordan_block (L,n)),i)) . i = (Jordan_block (L,n)) * (i,i) & f . i = f /. i ) by A1, A3, A9, MATRIX_1:def_8, PARTFUN1:def_6;
A14: [i,i] in Indices (Jordan_block (L,n)) by A1, A7, ZFMISC_1:87;
assume A15: i = 1 ; ::_thesis: (Col ((Jordan_block (L,n)),i)) "*" F = L * (F /. i)
now__::_thesis:_for_j_being_Nat_st_j_in_Seg_n_&_j_<>_i_holds_
(mlt_((Col_((Jordan_block_(L,n)),i)),f))_._j_=_0._K
let j be Nat; ::_thesis: ( j in Seg n & j <> i implies (mlt ((Col ((Jordan_block (L,n)),i)),f)) . j = 0. K )
assume that
A16: j in Seg n and
A17: j <> i ; ::_thesis: (mlt ((Col ((Jordan_block (L,n)),i)),f)) . j = 0. K
A18: f . j = f /. j by A9, A16, PARTFUN1:def_6;
1 <= j by A16, FINSEQ_1:1;
then A19: i < j + 1 by A15, NAT_1:13;
A20: [j,i] in Indices (Jordan_block (L,n)) by A1, A7, A16, ZFMISC_1:87;
(Col ((Jordan_block (L,n)),i)) . j = (Jordan_block (L,n)) * (j,i) by A3, A16, MATRIX_1:def_8
.= 0. K by A17, A19, A20, Def1 ;
hence (mlt ((Col ((Jordan_block (L,n)),i)),f)) . j = (0. K) * (f /. j) by A11, A16, A18, FVSUM_1:61
.= 0. K by VECTSP_1:6 ;
::_thesis: verum
end;
hence (Col ((Jordan_block (L,n)),i)) "*" F = (mlt ((Col ((Jordan_block (L,n)),i)),f)) . i by A1, A11, MATRIX_3:12
.= ((Jordan_block (L,n)) * (i,i)) * (f /. i) by A1, A11, A13, FVSUM_1:61
.= L * (F /. i) by A14, Def1 ;
::_thesis: verum
end;
A21: i1 <> i ;
assume i <> 1 ; ::_thesis: (Col ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i - 1))
then i1 + 1 > 0 + 1 by A2, XXREAL_0:1;
then i1 >= 1 by NAT_1:14;
then A22: i1 in Seg n by A6;
then A23: ( i1 + 1 = i & [i1,i] in Indices (Jordan_block (L,n)) ) by A1, A7, ZFMISC_1:87;
A24: now__::_thesis:_for_j_being_Nat_st_j_in_Seg_n_&_j_<>_i_&_j_<>_i1_holds_
(mlt_(Ci,f))_._j_=_0._K
let j be Nat; ::_thesis: ( j in Seg n & j <> i & j <> i1 implies (mlt (Ci,f)) . j = 0. K )
assume that
A25: j in Seg n and
A26: j <> i and
A27: j <> i1 ; ::_thesis: (mlt (Ci,f)) . j = 0. K
( [j,i] in Indices (Jordan_block (L,n)) & j + 1 <> i ) by A1, A7, A25, A27, ZFMISC_1:87;
then A28: 0. K = (Jordan_block (L,n)) * (j,i) by A26, Def1
.= Ci . j by A3, A25, MATRIX_1:def_8 ;
f . j = f /. j by A9, A25, PARTFUN1:def_6;
hence (mlt (Ci,f)) . j = (0. K) * (f /. j) by A25, A28, FVSUM_1:61
.= 0. K by VECTSP_1:6 ;
::_thesis: verum
end;
A29: f . i1 = f /. i1 by A9, A22, PARTFUN1:def_6;
A30: (Col ((Jordan_block (L,n)),i)) . i1 = (Jordan_block (L,n)) * (i1,i) by A3, A22, MATRIX_1:def_8;
(mlt (Ci,f)) /. i1 = (mlt (Ci,f)) . i1 by A11, A22, PARTFUN1:def_6
.= ((Jordan_block (L,n)) * (i1,i)) * (f /. i1) by A22, A30, A29, FVSUM_1:61
.= (1_ K) * (f /. i1) by A23, Def1
.= f /. i1 by VECTSP_1:def_4 ;
hence (Col ((Jordan_block (L,n)),i)) "*" F = (L * (F /. i)) + (F /. (i - 1)) by A1, A11, A22, A21, A12, A24, MATRIX15:7; ::_thesis: verum
end;
theorem :: MATRIXJ2:8
for n being Nat
for K being Field
for L being Element of K st L <> 0. K holds
ex M being Matrix of n,K st
( (Jordan_block (L,n)) ~ = M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i > j implies M * (i,j) = 0. K ) & ( i <= j implies M * (i,j) = - ((power K) . ((- (L ")),((j -' i) + 1))) ) ) ) )
proof
let n be Nat; ::_thesis: for K being Field
for L being Element of K st L <> 0. K holds
ex M being Matrix of n,K st
( (Jordan_block (L,n)) ~ = M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i > j implies M * (i,j) = 0. K ) & ( i <= j implies M * (i,j) = - ((power K) . ((- (L ")),((j -' i) + 1))) ) ) ) )
let K be Field; ::_thesis: for L being Element of K st L <> 0. K holds
ex M being Matrix of n,K st
( (Jordan_block (L,n)) ~ = M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i > j implies M * (i,j) = 0. K ) & ( i <= j implies M * (i,j) = - ((power K) . ((- (L ")),((j -' i) + 1))) ) ) ) )
let L be Element of K; ::_thesis: ( L <> 0. K implies ex M being Matrix of n,K st
( (Jordan_block (L,n)) ~ = M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i > j implies M * (i,j) = 0. K ) & ( i <= j implies M * (i,j) = - ((power K) . ((- (L ")),((j -' i) + 1))) ) ) ) ) )
reconsider N = n as Element of NAT by ORDINAL1:def_12;
defpred S1[ Nat, Nat, set ] means ( ( $1 > $2 implies $3 = 0. K ) & ( $1 <= $2 implies $3 = - ((power K) . ((- (L ")),(($2 -' $1) + 1))) ) );
A1: for i, j being Nat st [i,j] in [:(Seg N),(Seg N):] holds
ex x being Element of K st S1[i,j,x]
proof
let i, j be Nat; ::_thesis: ( [i,j] in [:(Seg N),(Seg N):] implies ex x being Element of K st S1[i,j,x] )
assume [i,j] in [:(Seg N),(Seg N):] ; ::_thesis: ex x being Element of K st S1[i,j,x]
percases ( i > j or i <= j ) ;
supposeA2: i > j ; ::_thesis: ex x being Element of K st S1[i,j,x]
take 0. K ; ::_thesis: S1[i,j, 0. K]
thus S1[i,j, 0. K] by A2; ::_thesis: verum
end;
supposeA3: i <= j ; ::_thesis: ex x being Element of K st S1[i,j,x]
take - ((power K) . ((- (L ")),((j -' i) + 1))) ; ::_thesis: S1[i,j, - ((power K) . ((- (L ")),((j -' i) + 1)))]
thus S1[i,j, - ((power K) . ((- (L ")),((j -' i) + 1)))] by A3; ::_thesis: verum
end;
end;
end;
consider M being Matrix of N,K such that
A4: for i, j being Nat st [i,j] in Indices M holds
S1[i,j,M * (i,j)] from MATRIX_1:sch_2(A1);
set ONE = 1. (K,n);
set J = Jordan_block (L,n);
A5: Indices (1. (K,n)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
assume A6: L <> 0. K ; ::_thesis: ex M being Matrix of n,K st
( (Jordan_block (L,n)) ~ = M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i > j implies M * (i,j) = 0. K ) & ( i <= j implies M * (i,j) = - ((power K) . ((- (L ")),((j -' i) + 1))) ) ) ) )
then A7: Jordan_block (L,n) is invertible by Th3;
reconsider M = M as Matrix of n,K ;
set MJ = M * (Jordan_block (L,n));
A8: ( Indices M = Indices (Jordan_block (L,n)) & Indices (Jordan_block (L,n)) = Indices (1. (K,n)) ) by MATRIX_1:26;
A9: width M = n by MATRIX_1:24;
A10: ( Indices (M * (Jordan_block (L,n))) = Indices (1. (K,n)) & len (Jordan_block (L,n)) = n ) by MATRIX_1:24, MATRIX_1:26;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(1._(K,n))_holds_
(1._(K,n))_*_(i,j)_=_(M_*_(Jordan_block_(L,n)))_*_(i,j)
let i, j be Nat; ::_thesis: ( [i,j] in Indices (1. (K,n)) implies (1. (K,n)) * (i,j) = (M * (Jordan_block (L,n))) * (i,j) )
assume A11: [i,j] in Indices (1. (K,n)) ; ::_thesis: (1. (K,n)) * (i,j) = (M * (Jordan_block (L,n))) * (i,j)
A12: i in Seg n by A5, A11, ZFMISC_1:87;
set LL = Line (M,i);
A13: (M * (Jordan_block (L,n))) * (i,j) = (Line (M,i)) "*" (Col ((Jordan_block (L,n)),j)) by A9, A10, A11, MATRIX_3:def_4
.= (Col ((Jordan_block (L,n)),j)) "*" (Line (M,i)) by FVSUM_1:90 ;
A14: j in Seg n by A5, A11, ZFMISC_1:87;
then A15: (Line (M,i)) . j = M * (i,j) by A9, MATRIX_1:def_7;
A16: dom (Line (M,i)) = Seg n by A9, FINSEQ_2:124;
then A17: (Line (M,i)) /. j = (Line (M,i)) . j by A14, PARTFUN1:def_6;
now__::_thesis:_(1._(K,n))_*_(i,j)_=_(M_*_(Jordan_block_(L,n)))_*_(i,j)
percases ( j = 1 or j <> 1 ) ;
supposeA18: j = 1 ; ::_thesis: (1. (K,n)) * (i,j) = (M * (Jordan_block (L,n))) * (i,j)
then A19: (M * (Jordan_block (L,n))) * (i,j) = L * (M * (i,j)) by A9, A14, A17, A15, A13, Th7;
now__::_thesis:_(M_*_(Jordan_block_(L,n)))_*_(i,j)_=_(1._(K,n))_*_(i,j)
percases ( i = j or i <> j ) ;
supposeA20: i = j ; ::_thesis: (M * (Jordan_block (L,n))) * (i,j) = (1. (K,n)) * (i,j)
hence (M * (Jordan_block (L,n))) * (i,j) = L * (- ((power K) . ((- (L ")),((i -' i) + 1)))) by A4, A8, A11, A19
.= L * (- ((power K) . ((- (L ")),(0 + 1)))) by XREAL_1:232
.= L * (- (- (L "))) by GROUP_1:50
.= L * (L ") by RLVECT_1:17
.= 1_ K by A6, VECTSP_1:def_10
.= (1. (K,n)) * (i,j) by A11, A20, MATRIX_1:def_11 ;
::_thesis: verum
end;
supposeA21: i <> j ; ::_thesis: (M * (Jordan_block (L,n))) * (i,j) = (1. (K,n)) * (i,j)
1 <= i by A12, FINSEQ_1:1;
then j < i by A18, A21, XXREAL_0:1;
hence (M * (Jordan_block (L,n))) * (i,j) = L * (0. K) by A4, A8, A11, A19
.= 0. K by VECTSP_1:6
.= (1. (K,n)) * (i,j) by A11, A21, MATRIX_1:def_11 ;
::_thesis: verum
end;
end;
end;
hence (1. (K,n)) * (i,j) = (M * (Jordan_block (L,n))) * (i,j) ; ::_thesis: verum
end;
supposeA22: j <> 1 ; ::_thesis: (M * (Jordan_block (L,n))) * (i,j) = (1. (K,n)) * (i,j)
A23: j >= 1 by A14, FINSEQ_1:1;
then reconsider j1 = j - 1 as Element of NAT by NAT_1:21;
( j1 <= j1 + 1 & j <= n ) by A14, FINSEQ_1:1, NAT_1:11;
then A24: n >= j1 by XXREAL_0:2;
j1 + 1 > 0 + 1 by A22, A23, XXREAL_0:1;
then j1 >= 1 by NAT_1:14;
then A25: j1 in Seg n by A24;
then A26: [i,j1] in Indices (1. (K,n)) by A5, A12, ZFMISC_1:87;
( (Line (M,i)) /. j1 = (Line (M,i)) . j1 & (Line (M,i)) . j1 = M * (i,j1) ) by A9, A16, A25, MATRIX_1:def_7, PARTFUN1:def_6;
then A27: (M * (Jordan_block (L,n))) * (i,j) = (L * (M * (i,j))) + (M * (i,j1)) by A9, A14, A17, A15, A13, A22, Th7;
now__::_thesis:_(M_*_(Jordan_block_(L,n)))_*_(i,j)_=_(1._(K,n))_*_(i,j)
percases ( i < j1 + 1 or i = j1 + 1 or i > j1 + 1 ) by XXREAL_0:1;
supposeA28: i < j1 + 1 ; ::_thesis: (M * (Jordan_block (L,n))) * (i,j) = (1. (K,n)) * (i,j)
set P = (power K) . ((- (L ")),((j1 -' i) + 1));
A29: j -' i = j - i by A28, XREAL_1:233;
A30: i <= j1 by A28, NAT_1:13;
then A31: j1 -' i = j1 - i by XREAL_1:233;
thus (M * (Jordan_block (L,n))) * (i,j) = (L * (M * (i,j))) + (- ((power K) . ((- (L ")),((j1 -' i) + 1)))) by A4, A8, A26, A27, A30
.= (L * (- ((power K) . ((- (L ")),((j -' i) + 1))))) + (- ((power K) . ((- (L ")),((j1 -' i) + 1)))) by A4, A8, A11, A28
.= (L * (- ((- (L ")) * ((power K) . ((- (L ")),((j1 -' i) + 1)))))) + (- ((power K) . ((- (L ")),((j1 -' i) + 1)))) by A31, A29, GROUP_1:def_7
.= (L * ((- (- (L "))) * ((power K) . ((- (L ")),((j1 -' i) + 1))))) + (- ((power K) . ((- (L ")),((j1 -' i) + 1)))) by VECTSP_1:9
.= (L * ((L ") * ((power K) . ((- (L ")),((j1 -' i) + 1))))) + (- ((power K) . ((- (L ")),((j1 -' i) + 1)))) by RLVECT_1:17
.= ((L * (L ")) * ((power K) . ((- (L ")),((j1 -' i) + 1)))) + (- ((power K) . ((- (L ")),((j1 -' i) + 1)))) by GROUP_1:def_3
.= ((1_ K) * ((power K) . ((- (L ")),((j1 -' i) + 1)))) + (- ((power K) . ((- (L ")),((j1 -' i) + 1)))) by A6, VECTSP_1:def_10
.= ((power K) . ((- (L ")),((j1 -' i) + 1))) + (- ((power K) . ((- (L ")),((j1 -' i) + 1)))) by VECTSP_1:def_6
.= 0. K by RLVECT_1:def_10
.= (1. (K,n)) * (i,j) by A11, A28, MATRIX_1:def_11 ; ::_thesis: verum
end;
supposeA32: i = j1 + 1 ; ::_thesis: (M * (Jordan_block (L,n))) * (i,j) = (1. (K,n)) * (i,j)
then i > j1 by NAT_1:13;
hence (M * (Jordan_block (L,n))) * (i,j) = (L * (M * (i,j))) + (0. K) by A4, A8, A26, A27
.= L * (M * (i,j)) by RLVECT_1:def_4
.= L * (- ((power K) . ((- (L ")),((i -' i) + 1)))) by A4, A8, A11, A32
.= L * (- ((power K) . ((- (L ")),(0 + 1)))) by XREAL_1:232
.= L * (- (- (L "))) by GROUP_1:50
.= L * (L ") by RLVECT_1:17
.= 1_ K by A6, VECTSP_1:def_10
.= (1. (K,n)) * (i,j) by A11, A32, MATRIX_1:def_11 ;
::_thesis: verum
end;
supposeA33: i > j1 + 1 ; ::_thesis: (M * (Jordan_block (L,n))) * (i,j) = (1. (K,n)) * (i,j)
then i > j1 by NAT_1:13;
hence (M * (Jordan_block (L,n))) * (i,j) = (L * (M * (i,j))) + (0. K) by A4, A8, A26, A27
.= L * (M * (i,j)) by RLVECT_1:def_4
.= L * (0. K) by A4, A8, A11, A33
.= 0. K by VECTSP_1:6
.= (1. (K,n)) * (i,j) by A11, A33, MATRIX_1:def_11 ;
::_thesis: verum
end;
end;
end;
hence (M * (Jordan_block (L,n))) * (i,j) = (1. (K,n)) * (i,j) ; ::_thesis: verum
end;
end;
end;
hence (1. (K,n)) * (i,j) = (M * (Jordan_block (L,n))) * (i,j) ; ::_thesis: verum
end;
then A34: 1. (K,n) = M * (Jordan_block (L,n)) by MATRIX_1:27;
set JM = (Jordan_block (L,n)) * M;
A35: len M = n by MATRIX_1:24;
take M ; ::_thesis: ( (Jordan_block (L,n)) ~ = M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i > j implies M * (i,j) = 0. K ) & ( i <= j implies M * (i,j) = - ((power K) . ((- (L ")),((j -' i) + 1))) ) ) ) )
A36: ( Indices ((Jordan_block (L,n)) * M) = Indices (1. (K,n)) & width (Jordan_block (L,n)) = n ) by MATRIX_1:24, MATRIX_1:26;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(1._(K,n))_holds_
(1._(K,n))_*_(i,j)_=_((Jordan_block_(L,n))_*_M)_*_(i,j)
let i, j be Nat; ::_thesis: ( [i,j] in Indices (1. (K,n)) implies (1. (K,n)) * (i,j) = ((Jordan_block (L,n)) * M) * (i,j) )
assume A37: [i,j] in Indices (1. (K,n)) ; ::_thesis: (1. (K,n)) * (i,j) = ((Jordan_block (L,n)) * M) * (i,j)
A38: i in Seg n by A5, A37, ZFMISC_1:87;
set i1 = i + 1;
A39: j in Seg n by A5, A37, ZFMISC_1:87;
A40: ((Jordan_block (L,n)) * M) * (i,j) = (Line ((Jordan_block (L,n)),i)) "*" (Col (M,j)) by A35, A36, A37, MATRIX_3:def_4;
set C = Col (M,j);
A41: dom M = Seg n by A35, FINSEQ_1:def_3;
then A42: (Col (M,j)) . i = M * (i,j) by A38, MATRIX_1:def_8;
A43: dom (Col (M,j)) = Seg n by A35, FINSEQ_2:124;
then A44: (Col (M,j)) /. i = (Col (M,j)) . i by A38, PARTFUN1:def_6;
now__::_thesis:_((Jordan_block_(L,n))_*_M)_*_(i,j)_=_(1._(K,n))_*_(i,j)
percases ( i = n or i <> n ) ;
supposeA45: i = n ; ::_thesis: ((Jordan_block (L,n)) * M) * (i,j) = (1. (K,n)) * (i,j)
then A46: ((Jordan_block (L,n)) * M) * (i,j) = L * (M * (i,j)) by A35, A38, A44, A42, A40, Th6;
now__::_thesis:_((Jordan_block_(L,n))_*_M)_*_(i,j)_=_(1._(K,n))_*_(i,j)
percases ( i > j or i <= j ) ;
supposeA47: i > j ; ::_thesis: ((Jordan_block (L,n)) * M) * (i,j) = (1. (K,n)) * (i,j)
hence ((Jordan_block (L,n)) * M) * (i,j) = L * (0. K) by A4, A8, A37, A46
.= 0. K by VECTSP_1:6
.= (1. (K,n)) * (i,j) by A37, A47, MATRIX_1:def_11 ;
::_thesis: verum
end;
supposeA48: i <= j ; ::_thesis: ((Jordan_block (L,n)) * M) * (i,j) = (1. (K,n)) * (i,j)
j <= n by A39, FINSEQ_1:1;
then A49: j = n by A45, A48, XXREAL_0:1;
hence ((Jordan_block (L,n)) * M) * (i,j) = L * (- ((power K) . ((- (L ")),((n -' n) + 1)))) by A4, A8, A37, A45, A46
.= L * (- ((power K) . ((- (L ")),(0 + 1)))) by XREAL_1:232
.= L * (- (- (L "))) by GROUP_1:50
.= L * (L ") by RLVECT_1:17
.= 1_ K by A6, VECTSP_1:def_10
.= (1. (K,n)) * (i,j) by A37, A45, A49, MATRIX_1:def_11 ;
::_thesis: verum
end;
end;
end;
hence ((Jordan_block (L,n)) * M) * (i,j) = (1. (K,n)) * (i,j) ; ::_thesis: verum
end;
supposeA50: i <> n ; ::_thesis: (1. (K,n)) * (i,j) = ((Jordan_block (L,n)) * M) * (i,j)
i <= n by A38, FINSEQ_1:1;
then i < n by A50, XXREAL_0:1;
then ( 1 <= i + 1 & i + 1 <= n ) by NAT_1:11, NAT_1:13;
then A51: i + 1 in Seg n ;
then A52: [(i + 1),j] in Indices M by A8, A5, A39, ZFMISC_1:87;
( (Col (M,j)) /. (i + 1) = (Col (M,j)) . (i + 1) & (Col (M,j)) . (i + 1) = M * ((i + 1),j) ) by A41, A43, A51, MATRIX_1:def_8, PARTFUN1:def_6;
then A53: ((Jordan_block (L,n)) * M) * (i,j) = (L * (M * (i,j))) + (M * ((i + 1),j)) by A35, A38, A44, A42, A40, A50, Th6;
now__::_thesis:_((Jordan_block_(L,n))_*_M)_*_(i,j)_=_(1._(K,n))_*_(i,j)
percases ( i > j or i = j or i < j ) by XXREAL_0:1;
supposeA54: i > j ; ::_thesis: ((Jordan_block (L,n)) * M) * (i,j) = (1. (K,n)) * (i,j)
then i + 1 > j by NAT_1:13;
hence ((Jordan_block (L,n)) * M) * (i,j) = (L * (M * (i,j))) + (0. K) by A4, A52, A53
.= L * (M * (i,j)) by RLVECT_1:def_4
.= L * (0. K) by A4, A8, A37, A54
.= 0. K by VECTSP_1:6
.= (1. (K,n)) * (i,j) by A37, A54, MATRIX_1:def_11 ;
::_thesis: verum
end;
supposeA55: i = j ; ::_thesis: ((Jordan_block (L,n)) * M) * (i,j) = (1. (K,n)) * (i,j)
then i + 1 > j by NAT_1:13;
hence ((Jordan_block (L,n)) * M) * (i,j) = (L * (M * (i,j))) + (0. K) by A4, A52, A53
.= L * (M * (i,i)) by A55, RLVECT_1:def_4
.= L * (- ((power K) . ((- (L ")),((i -' i) + 1)))) by A4, A8, A37, A55
.= L * (- ((power K) . ((- (L ")),(0 + 1)))) by XREAL_1:232
.= L * (- (- (L "))) by GROUP_1:50
.= L * (L ") by RLVECT_1:17
.= 1_ K by A6, VECTSP_1:def_10
.= (1. (K,n)) * (i,j) by A37, A55, MATRIX_1:def_11 ;
::_thesis: verum
end;
supposeA56: i < j ; ::_thesis: ((Jordan_block (L,n)) * M) * (i,j) = (1. (K,n)) * (i,j)
set P = (power K) . ((- (L ")),((j -' (i + 1)) + 1));
A57: j -' i = j - i by A56, XREAL_1:233;
A58: i + 1 <= j by A56, NAT_1:13;
then A59: j -' (i + 1) = j - (i + 1) by XREAL_1:233;
thus ((Jordan_block (L,n)) * M) * (i,j) = (L * (M * (i,j))) + (- ((power K) . ((- (L ")),((j -' (i + 1)) + 1)))) by A4, A52, A53, A58
.= (L * (- ((power K) . ((- (L ")),((j -' i) + 1))))) + (- ((power K) . ((- (L ")),((j -' (i + 1)) + 1)))) by A4, A8, A37, A56
.= (L * (- ((- (L ")) * ((power K) . ((- (L ")),((j -' (i + 1)) + 1)))))) + (- ((power K) . ((- (L ")),((j -' (i + 1)) + 1)))) by A59, A57, GROUP_1:def_7
.= (L * ((- (- (L "))) * ((power K) . ((- (L ")),((j -' (i + 1)) + 1))))) + (- ((power K) . ((- (L ")),((j -' (i + 1)) + 1)))) by VECTSP_1:9
.= (L * ((L ") * ((power K) . ((- (L ")),((j -' (i + 1)) + 1))))) + (- ((power K) . ((- (L ")),((j -' (i + 1)) + 1)))) by RLVECT_1:17
.= ((L * (L ")) * ((power K) . ((- (L ")),((j -' (i + 1)) + 1)))) + (- ((power K) . ((- (L ")),((j -' (i + 1)) + 1)))) by GROUP_1:def_3
.= ((1_ K) * ((power K) . ((- (L ")),((j -' (i + 1)) + 1)))) + (- ((power K) . ((- (L ")),((j -' (i + 1)) + 1)))) by A6, VECTSP_1:def_10
.= ((power K) . ((- (L ")),((j -' (i + 1)) + 1))) + (- ((power K) . ((- (L ")),((j -' (i + 1)) + 1)))) by VECTSP_1:def_6
.= 0. K by RLVECT_1:def_10
.= (1. (K,n)) * (i,j) by A37, A56, MATRIX_1:def_11 ; ::_thesis: verum
end;
end;
end;
hence (1. (K,n)) * (i,j) = ((Jordan_block (L,n)) * M) * (i,j) ; ::_thesis: verum
end;
end;
end;
hence (1. (K,n)) * (i,j) = ((Jordan_block (L,n)) * M) * (i,j) ; ::_thesis: verum
end;
then 1. (K,n) = (Jordan_block (L,n)) * M by MATRIX_1:27;
then M is_reverse_of Jordan_block (L,n) by A34, MATRIX_6:def_2;
hence ( (Jordan_block (L,n)) ~ = M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i > j implies M * (i,j) = 0. K ) & ( i <= j implies M * (i,j) = - ((power K) . ((- (L ")),((j -' i) + 1))) ) ) ) ) by A4, A7, MATRIX_6:def_4; ::_thesis: verum
end;
theorem Th9: :: MATRIXJ2:9
for n being Nat
for K being Field
for L, a being Element of K holds (Jordan_block (L,n)) + (a * (1. (K,n))) = Jordan_block ((L + a),n)
proof
let n be Nat; ::_thesis: for K being Field
for L, a being Element of K holds (Jordan_block (L,n)) + (a * (1. (K,n))) = Jordan_block ((L + a),n)
let K be Field; ::_thesis: for L, a being Element of K holds (Jordan_block (L,n)) + (a * (1. (K,n))) = Jordan_block ((L + a),n)
let L, a be Element of K; ::_thesis: (Jordan_block (L,n)) + (a * (1. (K,n))) = Jordan_block ((L + a),n)
set J = Jordan_block (L,n);
set Ja = Jordan_block ((L + a),n);
set ONE = 1. (K,n);
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(Jordan_block_((L_+_a),n))_holds_
(Jordan_block_((L_+_a),n))_*_(i,j)_=_((Jordan_block_(L,n))_+_(a_*_(1._(K,n))))_*_(i,j)
A1: Indices (Jordan_block (L,n)) = Indices (Jordan_block ((L + a),n)) by MATRIX_1:26;
let i, j be Nat; ::_thesis: ( [i,j] in Indices (Jordan_block ((L + a),n)) implies (Jordan_block ((L + a),n)) * (i,j) = ((Jordan_block (L,n)) + (a * (1. (K,n)))) * (i,j) )
assume A2: [i,j] in Indices (Jordan_block ((L + a),n)) ; ::_thesis: (Jordan_block ((L + a),n)) * (i,j) = ((Jordan_block (L,n)) + (a * (1. (K,n)))) * (i,j)
A3: Indices (Jordan_block (L,n)) = Indices (1. (K,n)) by MATRIX_1:26;
now__::_thesis:_(Jordan_block_((L_+_a),n))_*_(i,j)_=_((Jordan_block_(L,n))_+_(a_*_(1._(K,n))))_*_(i,j)
percases ( i = j or i + 1 = j or ( i <> j & i + 1 <> j ) ) ;
supposeA4: i = j ; ::_thesis: (Jordan_block ((L + a),n)) * (i,j) = ((Jordan_block (L,n)) + (a * (1. (K,n)))) * (i,j)
hence (Jordan_block ((L + a),n)) * (i,j) = L + a by A2, Def1
.= ((Jordan_block (L,n)) * (i,j)) + a by A2, A1, A4, Def1
.= ((Jordan_block (L,n)) * (i,j)) + (a * (1_ K)) by VECTSP_1:def_6
.= ((Jordan_block (L,n)) * (i,j)) + (a * ((1. (K,n)) * (i,j))) by A2, A1, A3, A4, MATRIX_1:def_11
.= ((Jordan_block (L,n)) * (i,j)) + ((a * (1. (K,n))) * (i,j)) by A2, A1, A3, MATRIX_3:def_5
.= ((Jordan_block (L,n)) + (a * (1. (K,n)))) * (i,j) by A2, A1, MATRIX_3:def_3 ;
::_thesis: verum
end;
supposeA5: i + 1 = j ; ::_thesis: (Jordan_block ((L + a),n)) * (i,j) = ((Jordan_block (L,n)) + (a * (1. (K,n)))) * (i,j)
then A6: i <> j ;
thus (Jordan_block ((L + a),n)) * (i,j) = 1_ K by A2, A5, Def1
.= (1_ K) + (0. K) by RLVECT_1:def_4
.= ((Jordan_block (L,n)) * (i,j)) + (0. K) by A2, A1, A5, Def1
.= ((Jordan_block (L,n)) * (i,j)) + (a * (0. K)) by VECTSP_1:6
.= ((Jordan_block (L,n)) * (i,j)) + (a * ((1. (K,n)) * (i,j))) by A2, A1, A3, A6, MATRIX_1:def_11
.= ((Jordan_block (L,n)) * (i,j)) + ((a * (1. (K,n))) * (i,j)) by A2, A1, A3, MATRIX_3:def_5
.= ((Jordan_block (L,n)) + (a * (1. (K,n)))) * (i,j) by A2, A1, MATRIX_3:def_3 ; ::_thesis: verum
end;
supposeA7: ( i <> j & i + 1 <> j ) ; ::_thesis: (Jordan_block ((L + a),n)) * (i,j) = ((Jordan_block (L,n)) + (a * (1. (K,n)))) * (i,j)
hence (Jordan_block ((L + a),n)) * (i,j) = 0. K by A2, Def1
.= (0. K) + (0. K) by RLVECT_1:def_4
.= ((Jordan_block (L,n)) * (i,j)) + (0. K) by A2, A1, A7, Def1
.= ((Jordan_block (L,n)) * (i,j)) + (a * (0. K)) by VECTSP_1:6
.= ((Jordan_block (L,n)) * (i,j)) + (a * ((1. (K,n)) * (i,j))) by A2, A1, A3, A7, MATRIX_1:def_11
.= ((Jordan_block (L,n)) * (i,j)) + ((a * (1. (K,n))) * (i,j)) by A2, A1, A3, MATRIX_3:def_5
.= ((Jordan_block (L,n)) + (a * (1. (K,n)))) * (i,j) by A2, A1, MATRIX_3:def_3 ;
::_thesis: verum
end;
end;
end;
hence (Jordan_block ((L + a),n)) * (i,j) = ((Jordan_block (L,n)) + (a * (1. (K,n)))) * (i,j) ; ::_thesis: verum
end;
hence (Jordan_block (L,n)) + (a * (1. (K,n))) = Jordan_block ((L + a),n) by MATRIX_1:27; ::_thesis: verum
end;
begin
definition
let K be Field;
let G be FinSequence of ( the carrier of K *) * ;
attrG is Jordan-block-yielding means :Def2: :: MATRIXJ2:def 2
for i being Nat st i in dom G holds
ex L being Element of K ex n being Nat st G . i = Jordan_block (L,n);
end;
:: deftheorem Def2 defines Jordan-block-yielding MATRIXJ2:def_2_:_
for K being Field
for G being FinSequence of ( the carrier of K *) * holds
( G is Jordan-block-yielding iff for i being Nat st i in dom G holds
ex L being Element of K ex n being Nat st G . i = Jordan_block (L,n) );
registration
let K be Field;
cluster Relation-like NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Jordan-block-yielding for FinSequence of ( the carrier of K *) * ;
existence
ex b1 being FinSequence of ( the carrier of K *) * st b1 is Jordan-block-yielding
proof
reconsider F = <*> (( the carrier of K *) *) as FinSequence of ( the carrier of K *) * ;
take F ; ::_thesis: F is Jordan-block-yielding
for i being Nat st i in dom F holds
ex L being Element of K ex n being Nat st F . i = Jordan_block (L,n) ;
hence F is Jordan-block-yielding by Def2; ::_thesis: verum
end;
end;
registration
let K be Field;
cluster Jordan-block-yielding -> Square-Matrix-yielding for FinSequence of ( the carrier of K *) * ;
coherence
for b1 being FinSequence of ( the carrier of K *) * st b1 is Jordan-block-yielding holds
b1 is Square-Matrix-yielding
proof
let F be FinSequence of ( the carrier of K *) * ; ::_thesis: ( F is Jordan-block-yielding implies F is Square-Matrix-yielding )
assume A1: F is Jordan-block-yielding ; ::_thesis: F is Square-Matrix-yielding
let i be Nat; :: according to MATRIXJ1:def_6 ::_thesis: ( not i in dom F or ex b1 being set st F . i is Matrix of b1,b1, the carrier of K )
assume i in dom F ; ::_thesis: ex b1 being set st F . i is Matrix of b1,b1, the carrier of K
then ex L being Element of K ex n being Nat st F . i = Jordan_block (L,n) by A1, Def2;
hence ex b1 being set st F . i is Matrix of b1,b1, the carrier of K ; ::_thesis: verum
end;
end;
definition
let K be Field;
mode FinSequence_of_Jordan_block of K is Jordan-block-yielding FinSequence of ( the carrier of K *) * ;
end;
Lm2: for K being Field holds {} is FinSequence_of_Jordan_block of K
proof
let K be Field; ::_thesis: {} is FinSequence_of_Jordan_block of K
reconsider F = <*> (( the carrier of K *) *) as FinSequence of ( the carrier of K *) * ;
for i being Nat st i in dom F holds
ex L being Element of K ex n being Nat st F . i = Jordan_block (L,n) ;
hence {} is FinSequence_of_Jordan_block of K by Def2; ::_thesis: verum
end;
definition
let K be Field;
let L be Element of K;
mode FinSequence_of_Jordan_block of L,K -> FinSequence_of_Jordan_block of K means :Def3: :: MATRIXJ2:def 3
for i being Nat st i in dom it holds
ex n being Nat st it . i = Jordan_block (L,n);
existence
ex b1 being FinSequence_of_Jordan_block of K st
for i being Nat st i in dom b1 holds
ex n being Nat st b1 . i = Jordan_block (L,n)
proof
reconsider F = <*> {} as FinSequence_of_Jordan_block of K by Lm2;
take F ; ::_thesis: for i being Nat st i in dom F holds
ex n being Nat st F . i = Jordan_block (L,n)
thus for i being Nat st i in dom F holds
ex n being Nat st F . i = Jordan_block (L,n) ; ::_thesis: verum
end;
end;
:: deftheorem Def3 defines FinSequence_of_Jordan_block MATRIXJ2:def_3_:_
for K being Field
for L being Element of K
for b3 being FinSequence_of_Jordan_block of K holds
( b3 is FinSequence_of_Jordan_block of L,K iff for i being Nat st i in dom b3 holds
ex n being Nat st b3 . i = Jordan_block (L,n) );
theorem Th10: :: MATRIXJ2:10
for K being Field
for L being Element of K holds {} is FinSequence_of_Jordan_block of L,K
proof
let K be Field; ::_thesis: for L being Element of K holds {} is FinSequence_of_Jordan_block of L,K
let L be Element of K; ::_thesis: {} is FinSequence_of_Jordan_block of L,K
reconsider F = <*> {} as FinSequence_of_Jordan_block of K by Lm2;
for i being Nat st i in dom F holds
ex n being Nat st F . i = Jordan_block (L,n) ;
hence {} is FinSequence_of_Jordan_block of L,K by Def3; ::_thesis: verum
end;
theorem Th11: :: MATRIXJ2:11
for n being Nat
for K being Field
for L being Element of K holds <*(Jordan_block (L,n))*> is FinSequence_of_Jordan_block of L,K
proof
let n be Nat; ::_thesis: for K being Field
for L being Element of K holds <*(Jordan_block (L,n))*> is FinSequence_of_Jordan_block of L,K
let K be Field; ::_thesis: for L being Element of K holds <*(Jordan_block (L,n))*> is FinSequence_of_Jordan_block of L,K
let L be Element of K; ::_thesis: <*(Jordan_block (L,n))*> is FinSequence_of_Jordan_block of L,K
now__::_thesis:_for_i_being_Nat_st_i_in_dom_<*(Jordan_block_(L,n))*>_holds_
ex_a_being_Element_of_K_ex_k_being_Nat_st_<*(Jordan_block_(L,n))*>_._i_=_Jordan_block_(a,k)
A1: dom <*(Jordan_block (L,n))*> = {1} by FINSEQ_1:2, FINSEQ_1:def_8;
let i be Nat; ::_thesis: ( i in dom <*(Jordan_block (L,n))*> implies ex a being Element of K ex k being Nat st <*(Jordan_block (L,n))*> . i = Jordan_block (a,k) )
assume i in dom <*(Jordan_block (L,n))*> ; ::_thesis: ex a being Element of K ex k being Nat st <*(Jordan_block (L,n))*> . i = Jordan_block (a,k)
then ( <*(Jordan_block (L,n))*> . 1 = Jordan_block (L,n) & i = 1 ) by A1, FINSEQ_1:def_8, TARSKI:def_1;
hence ex a being Element of K ex k being Nat st <*(Jordan_block (L,n))*> . i = Jordan_block (a,k) ; ::_thesis: verum
end;
then reconsider JJ = <*(Jordan_block (L,n))*> as FinSequence_of_Jordan_block of K by Def2;
now__::_thesis:_for_i_being_Nat_st_i_in_dom_JJ_holds_
ex_n_being_Nat_st_JJ_._i_=_Jordan_block_(L,n)
A2: dom JJ = {1} by FINSEQ_1:2, FINSEQ_1:def_8;
let i be Nat; ::_thesis: ( i in dom JJ implies ex n being Nat st JJ . i = Jordan_block (L,n) )
assume i in dom JJ ; ::_thesis: ex n being Nat st JJ . i = Jordan_block (L,n)
then ( JJ . 1 = Jordan_block (L,n) & i = 1 ) by A2, FINSEQ_1:def_8, TARSKI:def_1;
hence ex n being Nat st JJ . i = Jordan_block (L,n) ; ::_thesis: verum
end;
hence <*(Jordan_block (L,n))*> is FinSequence_of_Jordan_block of L,K by Def3; ::_thesis: verum
end;
registration
let K be Field;
let L be Element of K;
cluster Relation-like non-empty NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding Jordan-block-yielding for FinSequence_of_Jordan_block of L,K;
existence
ex b1 being FinSequence_of_Jordan_block of L,K st b1 is non-empty
proof
reconsider JJ = <*(Jordan_block (L,1))*> as FinSequence_of_Jordan_block of L,K by Th11;
take JJ ; ::_thesis: JJ is non-empty
now__::_thesis:_for_x_being_set_st_x_in_dom_JJ_holds_
not_JJ_._x_is_empty
A1: dom JJ = {1} by FINSEQ_1:2, FINSEQ_1:def_8;
let x be set ; ::_thesis: ( x in dom JJ implies not JJ . x is empty )
assume x in dom JJ ; ::_thesis: not JJ . x is empty
then ( JJ . 1 = Jordan_block (L,1) & x = 1 ) by A1, FINSEQ_1:def_8, TARSKI:def_1;
hence not JJ . x is empty by Def1; ::_thesis: verum
end;
hence JJ is non-empty by FUNCT_1:def_9; ::_thesis: verum
end;
end;
registration
let K be Field;
cluster Relation-like non-empty NAT -defined ( the carrier of K *) * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V201() Matrix-yielding Square-Matrix-yielding Jordan-block-yielding for FinSequence of ( the carrier of K *) * ;
existence
ex b1 being FinSequence_of_Jordan_block of K st b1 is non-empty
proof
set F = the non-empty FinSequence_of_Jordan_block of 1_ K,K;
take the non-empty FinSequence_of_Jordan_block of 1_ K,K ; ::_thesis: the non-empty FinSequence_of_Jordan_block of 1_ K,K is non-empty
thus the non-empty FinSequence_of_Jordan_block of 1_ K,K is non-empty ; ::_thesis: verum
end;
end;
theorem Th12: :: MATRIXJ2:12
for K being Field
for L, a being Element of K
for J being FinSequence_of_Jordan_block of L,K holds J (+) (mlt (((len J) |-> a),(1. (K,(Len J))))) is FinSequence_of_Jordan_block of L + a,K
proof
let K be Field; ::_thesis: for L, a being Element of K
for J being FinSequence_of_Jordan_block of L,K holds J (+) (mlt (((len J) |-> a),(1. (K,(Len J))))) is FinSequence_of_Jordan_block of L + a,K
let L, a be Element of K; ::_thesis: for J being FinSequence_of_Jordan_block of L,K holds J (+) (mlt (((len J) |-> a),(1. (K,(Len J))))) is FinSequence_of_Jordan_block of L + a,K
let J be FinSequence_of_Jordan_block of L,K; ::_thesis: J (+) (mlt (((len J) |-> a),(1. (K,(Len J))))) is FinSequence_of_Jordan_block of L + a,K
set M = mlt (((len J) |-> a),(1. (K,(Len J))));
A1: for i being Nat st i in dom (J (+) (mlt (((len J) |-> a),(1. (K,(Len J)))))) holds
ex n being Nat st (J (+) (mlt (((len J) |-> a),(1. (K,(Len J)))))) . i = Jordan_block ((L + a),n)
proof
A2: dom (mlt (((len J) |-> a),(1. (K,(Len J))))) = dom (1. (K,(Len J))) by MATRIXJ1:def_9;
A3: dom J = Seg (len J) by FINSEQ_1:def_3;
A4: dom (1. (K,(Len J))) = dom (Len J) by MATRIXJ1:def_8;
let i be Nat; ::_thesis: ( i in dom (J (+) (mlt (((len J) |-> a),(1. (K,(Len J)))))) implies ex n being Nat st (J (+) (mlt (((len J) |-> a),(1. (K,(Len J)))))) . i = Jordan_block ((L + a),n) )
assume A5: i in dom (J (+) (mlt (((len J) |-> a),(1. (K,(Len J)))))) ; ::_thesis: ex n being Nat st (J (+) (mlt (((len J) |-> a),(1. (K,(Len J)))))) . i = Jordan_block ((L + a),n)
A6: i in dom J by A5, MATRIXJ1:def_10;
then consider n being Nat such that
A7: J . i = Jordan_block (L,n) by Def3;
take n ; ::_thesis: (J (+) (mlt (((len J) |-> a),(1. (K,(Len J)))))) . i = Jordan_block ((L + a),n)
A8: len (J . i) = n by A7, MATRIX_1:24;
A9: dom (Len J) = dom J by MATRIXJ1:def_3;
then A10: (Len J) . i = len (J . i) by A6, MATRIXJ1:def_3;
len ((len J) |-> a) = len J by CARD_1:def_7;
then dom ((len J) |-> a) = dom J by FINSEQ_3:29;
then ((len J) |-> a) /. i = ((len J) |-> a) . i by A6, PARTFUN1:def_6
.= a by A6, A3, FINSEQ_2:57 ;
then (mlt (((len J) |-> a),(1. (K,(Len J))))) . i = a * ((1. (K,(Len J))) . i) by A6, A2, A4, A9, MATRIXJ1:def_9
.= a * (1. (K,n)) by A6, A4, A9, A10, A8, MATRIXJ1:def_8 ;
hence (J (+) (mlt (((len J) |-> a),(1. (K,(Len J)))))) . i = (Jordan_block (L,n)) + (a * (1. (K,n))) by A5, A7, MATRIXJ1:def_10
.= Jordan_block ((L + a),n) by Th9 ;
::_thesis: verum
end;
J (+) (mlt (((len J) |-> a),(1. (K,(Len J))))) is Jordan-block-yielding
proof
let i be Nat; :: according to MATRIXJ2:def_2 ::_thesis: ( i in dom (J (+) (mlt (((len J) |-> a),(1. (K,(Len J)))))) implies ex L being Element of K ex n being Nat st (J (+) (mlt (((len J) |-> a),(1. (K,(Len J)))))) . i = Jordan_block (L,n) )
assume i in dom (J (+) (mlt (((len J) |-> a),(1. (K,(Len J)))))) ; ::_thesis: ex L being Element of K ex n being Nat st (J (+) (mlt (((len J) |-> a),(1. (K,(Len J)))))) . i = Jordan_block (L,n)
then ex n being Nat st (J (+) (mlt (((len J) |-> a),(1. (K,(Len J)))))) . i = Jordan_block ((L + a),n) by A1;
hence ex L being Element of K ex n being Nat st (J (+) (mlt (((len J) |-> a),(1. (K,(Len J)))))) . i = Jordan_block (L,n) ; ::_thesis: verum
end;
then reconsider JM = J (+) (mlt (((len J) |-> a),(1. (K,(Len J))))) as FinSequence_of_Jordan_block of K ;
JM is FinSequence_of_Jordan_block of L + a,K
proof
let i be Nat; :: according to MATRIXJ2:def_3 ::_thesis: ( i in dom JM implies ex n being Nat st JM . i = Jordan_block ((L + a),n) )
assume i in dom JM ; ::_thesis: ex n being Nat st JM . i = Jordan_block ((L + a),n)
hence ex n being Nat st JM . i = Jordan_block ((L + a),n) by A1; ::_thesis: verum
end;
hence J (+) (mlt (((len J) |-> a),(1. (K,(Len J))))) is FinSequence_of_Jordan_block of L + a,K ; ::_thesis: verum
end;
definition
let K be Field;
let J1, J2 be FinSequence_of_Jordan_block of K;
:: original: ^
redefine funcJ1 ^ J2 -> FinSequence_of_Jordan_block of K;
coherence
J1 ^ J2 is FinSequence_of_Jordan_block of K
proof
J1 ^ J2 is Jordan-block-yielding
proof
let i be Nat; :: according to MATRIXJ2:def_2 ::_thesis: ( i in dom (J1 ^ J2) implies ex L being Element of K ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n) )
assume A1: i in dom (J1 ^ J2) ; ::_thesis: ex L being Element of K ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n)
percases ( i in dom J1 or ex n being Nat st
( n in dom J2 & i = (len J1) + n ) ) by A1, FINSEQ_1:25;
supposeA2: i in dom J1 ; ::_thesis: ex L being Element of K ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n)
then (J1 ^ J2) . i = J1 . i by FINSEQ_1:def_7;
hence ex L being Element of K ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n) by A2, Def2; ::_thesis: verum
end;
suppose ex n being Nat st
( n in dom J2 & i = (len J1) + n ) ; ::_thesis: ex L being Element of K ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n)
then consider k being Nat such that
A3: k in dom J2 and
A4: i = (len J1) + k ;
(J1 ^ J2) . i = J2 . k by A3, A4, FINSEQ_1:def_7;
hence ex L being Element of K ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n) by A3, Def2; ::_thesis: verum
end;
end;
end;
hence J1 ^ J2 is FinSequence_of_Jordan_block of K ; ::_thesis: verum
end;
end;
definition
let K be Field;
let n be Nat;
let J be FinSequence_of_Jordan_block of K;
:: original: |
redefine funcJ | n -> FinSequence_of_Jordan_block of K;
coherence
J | n is FinSequence_of_Jordan_block of K
proof
now__::_thesis:_for_i_being_Nat_st_i_in_dom_(J_|_n)_holds_
ex_L_being_Element_of_K_ex_k_being_Nat_st_(J_|_n)_._i_=_Jordan_block_(L,k)
let i be Nat; ::_thesis: ( i in dom (J | n) implies ex L being Element of K ex k being Nat st (J | n) . i = Jordan_block (L,k) )
assume i in dom (J | n) ; ::_thesis: ex L being Element of K ex k being Nat st (J | n) . i = Jordan_block (L,k)
then ( i in dom J & (J | n) . i = J . i ) by FUNCT_1:47, RELAT_1:57;
hence ex L being Element of K ex k being Nat st (J | n) . i = Jordan_block (L,k) by Def2; ::_thesis: verum
end;
hence J | n is FinSequence_of_Jordan_block of K by Def2; ::_thesis: verum
end;
:: original: /^
redefine funcJ /^ n -> FinSequence_of_Jordan_block of K;
coherence
J /^ n is FinSequence_of_Jordan_block of K
proof
now__::_thesis:_for_i_being_Nat_st_i_in_dom_(J_/^_n)_holds_
ex_L_being_Element_of_K_ex_k_being_Nat_st_(J_/^_n)_._i_=_Jordan_block_(L,k)
let i be Nat; ::_thesis: ( i in dom (J /^ n) implies ex L being Element of K ex k being Nat st (J /^ n) . i = Jordan_block (L,k) )
assume A1: i in dom (J /^ n) ; ::_thesis: ex L being Element of K ex k being Nat st (J /^ n) . i = Jordan_block (L,k)
i + n in dom J by A1, FINSEQ_5:26;
then A2: ( J . (n + i) = J /. (n + i) & ex L being Element of K ex k being Nat st J . (n + i) = Jordan_block (L,k) ) by Def2, PARTFUN1:def_6;
(J /^ n) . i = (J /^ n) /. i by A1, PARTFUN1:def_6;
hence ex L being Element of K ex k being Nat st (J /^ n) . i = Jordan_block (L,k) by A1, A2, FINSEQ_5:27; ::_thesis: verum
end;
hence J /^ n is FinSequence_of_Jordan_block of K by Def2; ::_thesis: verum
end;
end;
definition
let K be Field;
let L be Element of K;
let J1, J2 be FinSequence_of_Jordan_block of L,K;
:: original: ^
redefine funcJ1 ^ J2 -> FinSequence_of_Jordan_block of L,K;
coherence
J1 ^ J2 is FinSequence_of_Jordan_block of L,K
proof
J1 ^ J2 is FinSequence_of_Jordan_block of L,K
proof
let i be Nat; :: according to MATRIXJ2:def_3 ::_thesis: ( i in dom (J1 ^ J2) implies ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n) )
assume A1: i in dom (J1 ^ J2) ; ::_thesis: ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n)
percases ( i in dom J1 or ex n being Nat st
( n in dom J2 & i = (len J1) + n ) ) by A1, FINSEQ_1:25;
supposeA2: i in dom J1 ; ::_thesis: ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n)
then (J1 ^ J2) . i = J1 . i by FINSEQ_1:def_7;
hence ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n) by A2, Def3; ::_thesis: verum
end;
suppose ex n being Nat st
( n in dom J2 & i = (len J1) + n ) ; ::_thesis: ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n)
then consider k being Nat such that
A3: k in dom J2 and
A4: i = (len J1) + k ;
(J1 ^ J2) . i = J2 . k by A3, A4, FINSEQ_1:def_7;
hence ex n being Nat st (J1 ^ J2) . i = Jordan_block (L,n) by A3, Def3; ::_thesis: verum
end;
end;
end;
hence J1 ^ J2 is FinSequence_of_Jordan_block of L,K ; ::_thesis: verum
end;
end;
definition
let K be Field;
let L be Element of K;
let n be Nat;
let J be FinSequence_of_Jordan_block of L,K;
:: original: |
redefine funcJ | n -> FinSequence_of_Jordan_block of L,K;
coherence
J | n is FinSequence_of_Jordan_block of L,K
proof
now__::_thesis:_for_i_being_Nat_st_i_in_dom_(J_|_n)_holds_
ex_k_being_Nat_st_(J_|_n)_._i_=_Jordan_block_(L,k)
let i be Nat; ::_thesis: ( i in dom (J | n) implies ex k being Nat st (J | n) . i = Jordan_block (L,k) )
assume i in dom (J | n) ; ::_thesis: ex k being Nat st (J | n) . i = Jordan_block (L,k)
then ( i in dom J & (J | n) . i = J . i ) by FUNCT_1:47, RELAT_1:57;
hence ex k being Nat st (J | n) . i = Jordan_block (L,k) by Def3; ::_thesis: verum
end;
hence J | n is FinSequence_of_Jordan_block of L,K by Def3; ::_thesis: verum
end;
:: original: /^
redefine funcJ /^ n -> FinSequence_of_Jordan_block of L,K;
coherence
J /^ n is FinSequence_of_Jordan_block of L,K
proof
now__::_thesis:_for_i_being_Nat_st_i_in_dom_(J_/^_n)_holds_
ex_k_being_Nat_st_(J_/^_n)_._i_=_Jordan_block_(L,k)
let i be Nat; ::_thesis: ( i in dom (J /^ n) implies ex k being Nat st (J /^ n) . i = Jordan_block (L,k) )
assume A1: i in dom (J /^ n) ; ::_thesis: ex k being Nat st (J /^ n) . i = Jordan_block (L,k)
i + n in dom J by A1, FINSEQ_5:26;
then A2: ( J . (n + i) = J /. (n + i) & ex k being Nat st J . (n + i) = Jordan_block (L,k) ) by Def3, PARTFUN1:def_6;
(J /^ n) . i = (J /^ n) /. i by A1, PARTFUN1:def_6;
hence ex k being Nat st (J /^ n) . i = Jordan_block (L,k) by A1, A2, FINSEQ_5:27; ::_thesis: verum
end;
hence J /^ n is FinSequence_of_Jordan_block of L,K by Def3; ::_thesis: verum
end;
end;
begin
definition
let K be doubleLoopStr ;
let V be non empty VectSpStr over K;
let f be Function of V,V;
attrf is nilpotent means :Def4: :: MATRIXJ2:def 4
ex n being Nat st
for v being Vector of V holds (f |^ n) . v = 0. V;
end;
:: deftheorem Def4 defines nilpotent MATRIXJ2:def_4_:_
for K being doubleLoopStr
for V being non empty VectSpStr over K
for f being Function of V,V holds
( f is nilpotent iff ex n being Nat st
for v being Vector of V holds (f |^ n) . v = 0. V );
theorem Th13: :: MATRIXJ2:13
for K being doubleLoopStr
for V being non empty VectSpStr over K
for f being Function of V,V holds
( f is nilpotent iff ex n being Nat st f |^ n = ZeroMap (V,V) )
proof
let K be doubleLoopStr ; ::_thesis: for V being non empty VectSpStr over K
for f being Function of V,V holds
( f is nilpotent iff ex n being Nat st f |^ n = ZeroMap (V,V) )
let V be non empty VectSpStr over K; ::_thesis: for f being Function of V,V holds
( f is nilpotent iff ex n being Nat st f |^ n = ZeroMap (V,V) )
let f be Function of V,V; ::_thesis: ( f is nilpotent iff ex n being Nat st f |^ n = ZeroMap (V,V) )
hereby ::_thesis: ( ex n being Nat st f |^ n = ZeroMap (V,V) implies f is nilpotent )
assume f is nilpotent ; ::_thesis: ex n being Nat st f |^ n = ZeroMap (V,V)
then consider n being Nat such that
A1: for v being Vector of V holds (f |^ n) . v = 0. V by Def4;
now__::_thesis:_for_x_being_set_st_x_in_dom_(f_|^_n)_holds_
(f_|^_n)_._x_=_0._V
let x be set ; ::_thesis: ( x in dom (f |^ n) implies (f |^ n) . x = 0. V )
assume x in dom (f |^ n) ; ::_thesis: (f |^ n) . x = 0. V
then reconsider v = x as Vector of V by FUNCT_2:def_1;
thus (f |^ n) . x = (f |^ n) . v
.= 0. V by A1 ; ::_thesis: verum
end;
then f |^ n = (dom (f |^ n)) --> (0. V) by FUNCOP_1:11
.= the carrier of V --> (0. V) by FUNCT_2:def_1
.= ZeroMap (V,V) by GRCAT_1:def_7 ;
hence ex n being Nat st f |^ n = ZeroMap (V,V) ; ::_thesis: verum
end;
given n being Nat such that A2: f |^ n = ZeroMap (V,V) ; ::_thesis: f is nilpotent
take n ; :: according to MATRIXJ2:def_4 ::_thesis: for v being Vector of V holds (f |^ n) . v = 0. V
let v be Vector of V; ::_thesis: (f |^ n) . v = 0. V
thus (f |^ n) . v = ( the carrier of V --> (0. V)) . v by A2, GRCAT_1:def_7
.= 0. V by FUNCOP_1:7 ; ::_thesis: verum
end;
registration
let K be doubleLoopStr ;
let V be non empty VectSpStr over K;
cluster Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) nilpotent for Element of bool [: the carrier of V, the carrier of V:];
existence
ex b1 being Function of V,V st b1 is nilpotent
proof
take F = ZeroMap (V,V); ::_thesis: F is nilpotent
F |^ 1 = F by VECTSP11:19;
hence F is nilpotent by Th13; ::_thesis: verum
end;
end;
registration
let R be Ring;
let V be LeftMod of R;
cluster Relation-like the carrier of V -defined the carrier of V -valued Function-like non empty total V18( the carrier of V, the carrier of V) additive homogeneous nilpotent for Element of bool [: the carrier of V, the carrier of V:];
existence
ex b1 being Function of V,V st
( b1 is nilpotent & b1 is additive & b1 is homogeneous )
proof
take F = ZeroMap (V,V); ::_thesis: ( F is nilpotent & F is additive & F is homogeneous )
F |^ 1 = F by VECTSP11:19;
hence ( F is nilpotent & F is additive & F is homogeneous ) by Th13; ::_thesis: verum
end;
end;
theorem Th14: :: MATRIXJ2:14
for n being Nat
for K being Field
for V being VectSp of K
for f being linear-transformation of V,V holds f | (ker (f |^ n)) is nilpotent linear-transformation of (ker (f |^ n)),(ker (f |^ n))
proof
let n be Nat; ::_thesis: for K being Field
for V being VectSp of K
for f being linear-transformation of V,V holds f | (ker (f |^ n)) is nilpotent linear-transformation of (ker (f |^ n)),(ker (f |^ n))
let K be Field; ::_thesis: for V being VectSp of K
for f being linear-transformation of V,V holds f | (ker (f |^ n)) is nilpotent linear-transformation of (ker (f |^ n)),(ker (f |^ n))
let V be VectSp of K; ::_thesis: for f being linear-transformation of V,V holds f | (ker (f |^ n)) is nilpotent linear-transformation of (ker (f |^ n)),(ker (f |^ n))
let f be linear-transformation of V,V; ::_thesis: f | (ker (f |^ n)) is nilpotent linear-transformation of (ker (f |^ n)),(ker (f |^ n))
set KER = ker (f |^ n);
reconsider fK = f | (ker (f |^ n)) as linear-transformation of (ker (f |^ n)),(ker (f |^ n)) by VECTSP11:28;
now__::_thesis:_for_v_being_Vector_of_(ker_(f_|^_n))_holds_(fK_|^_n)_._v_=_0._(ker_(f_|^_n))
let v be Vector of (ker (f |^ n)); ::_thesis: (fK |^ n) . v = 0. (ker (f |^ n))
reconsider v1 = v as Vector of V by VECTSP_4:10;
A1: v1 in ker (f |^ n) by STRUCT_0:def_5;
thus (fK |^ n) . v = ((f |^ n) | (ker (f |^ n))) . v by VECTSP11:22
.= (f |^ n) . v1 by FUNCT_1:49
.= 0. V by A1, RANKNULL:10
.= 0. (ker (f |^ n)) by VECTSP_4:11 ; ::_thesis: verum
end;
hence f | (ker (f |^ n)) is nilpotent linear-transformation of (ker (f |^ n)),(ker (f |^ n)) by Def4; ::_thesis: verum
end;
definition
let K be doubleLoopStr ;
let V be non empty VectSpStr over K;
let f be nilpotent Function of V,V;
func degree_of_nilpotent f -> Nat means :Def5: :: MATRIXJ2:def 5
( f |^ it = ZeroMap (V,V) & ( for k being Nat st f |^ k = ZeroMap (V,V) holds
it <= k ) );
existence
ex b1 being Nat st
( f |^ b1 = ZeroMap (V,V) & ( for k being Nat st f |^ k = ZeroMap (V,V) holds
b1 <= k ) )
proof
defpred S1[ Nat] means f |^ $1 = ZeroMap (V,V);
A1: ex n being Nat st S1[n] by Th13;
consider n being Nat such that
A2: ( S1[n] & ( for k being Nat st S1[k] holds
n <= k ) ) from NAT_1:sch_5(A1);
take n ; ::_thesis: ( f |^ n = ZeroMap (V,V) & ( for k being Nat st f |^ k = ZeroMap (V,V) holds
n <= k ) )
thus ( f |^ n = ZeroMap (V,V) & ( for k being Nat st f |^ k = ZeroMap (V,V) holds
n <= k ) ) by A2; ::_thesis: verum
end;
uniqueness
for b1, b2 being Nat st f |^ b1 = ZeroMap (V,V) & ( for k being Nat st f |^ k = ZeroMap (V,V) holds
b1 <= k ) & f |^ b2 = ZeroMap (V,V) & ( for k being Nat st f |^ k = ZeroMap (V,V) holds
b2 <= k ) holds
b1 = b2
proof
let i, j be Nat; ::_thesis: ( f |^ i = ZeroMap (V,V) & ( for k being Nat st f |^ k = ZeroMap (V,V) holds
i <= k ) & f |^ j = ZeroMap (V,V) & ( for k being Nat st f |^ k = ZeroMap (V,V) holds
j <= k ) implies i = j )
assume ( f |^ i = ZeroMap (V,V) & ( for k being Nat st f |^ k = ZeroMap (V,V) holds
i <= k ) & f |^ j = ZeroMap (V,V) & ( for k being Nat st f |^ k = ZeroMap (V,V) holds
j <= k ) ) ; ::_thesis: i = j
then ( i <= j & j <= i ) ;
hence i = j by XXREAL_0:1; ::_thesis: verum
end;
end;
:: deftheorem Def5 defines degree_of_nilpotent MATRIXJ2:def_5_:_
for K being doubleLoopStr
for V being non empty VectSpStr over K
for f being nilpotent Function of V,V
for b4 being Nat holds
( b4 = degree_of_nilpotent f iff ( f |^ b4 = ZeroMap (V,V) & ( for k being Nat st f |^ k = ZeroMap (V,V) holds
b4 <= k ) ) );
notation
let K be doubleLoopStr ;
let V be non empty VectSpStr over K;
let f be nilpotent Function of V,V;
synonym deg f for degree_of_nilpotent f;
end;
theorem Th15: :: MATRIXJ2:15
for K being doubleLoopStr
for V being non empty VectSpStr over K
for f being nilpotent Function of V,V holds
( deg f = 0 iff [#] V = {(0. V)} )
proof
let K be doubleLoopStr ; ::_thesis: for V being non empty VectSpStr over K
for f being nilpotent Function of V,V holds
( deg f = 0 iff [#] V = {(0. V)} )
let V be non empty VectSpStr over K; ::_thesis: for f being nilpotent Function of V,V holds
( deg f = 0 iff [#] V = {(0. V)} )
let f be nilpotent Function of V,V; ::_thesis: ( deg f = 0 iff [#] V = {(0. V)} )
hereby ::_thesis: ( [#] V = {(0. V)} implies deg f = 0 )
assume A1: deg f = 0 ; ::_thesis: [#] V = {(0. V)}
[#] V c= {(0. V)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [#] V or x in {(0. V)} )
assume A2: x in [#] V ; ::_thesis: x in {(0. V)}
id V = f |^ 0 by VECTSP11:18
.= ZeroMap (V,V) by A1, Def5
.= the carrier of V --> (0. V) by GRCAT_1:def_7 ;
then x = ( the carrier of V --> (0. V)) . x by A2, FUNCT_1:18
.= 0. V by A2, FUNCOP_1:7 ;
hence x in {(0. V)} by TARSKI:def_1; ::_thesis: verum
end;
hence [#] V = {(0. V)} by ZFMISC_1:33; ::_thesis: verum
end;
assume A3: [#] V = {(0. V)} ; ::_thesis: deg f = 0
now__::_thesis:_for_x_being_set_st_x_in_dom_(f_|^_0)_holds_
(f_|^_0)_._x_=_0._V
let x be set ; ::_thesis: ( x in dom (f |^ 0) implies (f |^ 0) . x = 0. V )
assume x in dom (f |^ 0) ; ::_thesis: (f |^ 0) . x = 0. V
then reconsider v = x as Vector of V by FUNCT_2:def_1;
thus (f |^ 0) . x = (id V) . v by VECTSP11:18
.= 0. V by A3, TARSKI:def_1 ; ::_thesis: verum
end;
then f |^ 0 = (dom (f |^ 0)) --> (0. V) by FUNCOP_1:11
.= the carrier of V --> (0. V) by FUNCT_2:def_1
.= ZeroMap (V,V) by GRCAT_1:def_7 ;
hence deg f = 0 by Def5; ::_thesis: verum
end;
theorem Th16: :: MATRIXJ2:16
for K being doubleLoopStr
for V being non empty VectSpStr over K
for f being nilpotent Function of V,V ex v being Vector of V st
for i being Nat st i < deg f holds
(f |^ i) . v <> 0. V
proof
let K be doubleLoopStr ; ::_thesis: for V being non empty VectSpStr over K
for f being nilpotent Function of V,V ex v being Vector of V st
for i being Nat st i < deg f holds
(f |^ i) . v <> 0. V
let V be non empty VectSpStr over K; ::_thesis: for f being nilpotent Function of V,V ex v being Vector of V st
for i being Nat st i < deg f holds
(f |^ i) . v <> 0. V
let f be nilpotent Function of V,V; ::_thesis: ex v being Vector of V st
for i being Nat st i < deg f holds
(f |^ i) . v <> 0. V
set D = deg f;
defpred S1[ Nat] means ( 0 < $1 & $1 < deg f & (f |^ $1) . (0. V) = 0. V );
assume A1: for v being Vector of V ex i being Nat st
( i < deg f & (f |^ i) . v = 0. V ) ; ::_thesis: contradiction
then ex i being Nat st
( i < deg f & (f |^ i) . (0. V) = 0. V ) ;
then [#] V <> {(0. V)} by Th15;
then consider v being set such that
A2: v in [#] V and
A3: v <> 0. V by ZFMISC_1:35;
reconsider v = v as Vector of V by A2;
consider j being Nat such that
A4: j < deg f and
A5: (f |^ j) . v = 0. V by A1;
A6: j - j < (deg f) - j by A4, XREAL_1:9;
j > 0
proof
assume j <= 0 ; ::_thesis: contradiction
then j = 0 ;
then 0. V = (id V) . v by A5, VECTSP11:18
.= v by FUNCT_1:18 ;
hence contradiction by A3; ::_thesis: verum
end;
then A7: (deg f) - j < (deg f) - 0 by XREAL_1:10;
A8: dom (f |^ j) = [#] V by FUNCT_2:def_1;
A9: (deg f) - j = (deg f) -' j by A4, XREAL_1:233;
then A10: deg f = ((deg f) -' j) + j ;
A11: f |^ (deg f) = ZeroMap (V,V) by Def5
.= the carrier of V --> (0. V) by GRCAT_1:def_7 ;
then 0. V = (f |^ (deg f)) . v by FUNCOP_1:7
.= ((f |^ ((deg f) -' j)) * (f |^ j)) . v by A10, VECTSP11:20
.= (f |^ ((deg f) -' j)) . (0. V) by A5, A8, FUNCT_1:13 ;
then A12: ex j being Nat st S1[j] by A9, A6, A7;
consider m being Nat such that
A13: ( S1[m] & ( for n being Nat st S1[n] holds
m <= n ) ) from NAT_1:sch_5(A12);
A14: (deg f) -' m = (deg f) - m by A13, XREAL_1:233;
A15: now__::_thesis:_for_x_being_set_st_x_in_dom_(f_|^_((deg_f)_-'_m))_holds_
(f_|^_((deg_f)_-'_m))_._x_=_0._V
let x be set ; ::_thesis: ( x in dom (f |^ ((deg f) -' m)) implies (f |^ ((deg f) -' m)) . x = 0. V )
assume x in dom (f |^ ((deg f) -' m)) ; ::_thesis: (f |^ ((deg f) -' m)) . x = 0. V
then reconsider X = x as Vector of V by FUNCT_2:def_1;
consider k being Nat such that
A16: k < deg f and
A17: (f |^ k) . X = 0. V by A1;
defpred S2[ Nat] means ( $1 <= deg f & ex i being Nat st $1 = k + (i * m) );
k = k + (0 * m) ;
then A18: ex k being Nat st S2[k] by A16;
A19: for i being Nat st S2[i] holds
i <= deg f ;
consider MAX being Nat such that
A20: S2[MAX] and
A21: for k being Nat st S2[k] holds
k <= MAX from NAT_1:sch_6(A19, A18);
consider I being Nat such that
A22: MAX = k + (I * m) by A20;
now__::_thesis:_(f_|^_((deg_f)_-'_m))_._X_=_0._V
percases ( MAX = deg f or MAX < deg f ) by A20, XXREAL_0:1;
supposeA23: MAX = deg f ; ::_thesis: (f |^ ((deg f) -' m)) . X = 0. V
then I <> 0 by A16, A22;
then reconsider I1 = I - 1 as Nat by NAT_1:20;
(deg f) -' m = k + (I1 * m) by A14, A22, A23;
hence (f |^ ((deg f) -' m)) . X = 0. V by A13, A17, VECTSP11:21; ::_thesis: verum
end;
supposeA24: MAX < deg f ; ::_thesis: (f |^ ((deg f) -' m)) . X = 0. V
MAX <> 0
proof
assume A25: MAX = 0 ; ::_thesis: contradiction
then k = 0 by A22;
then k + (1 * m) < deg f by A13;
hence contradiction by A13, A21, A25; ::_thesis: verum
end;
then A26: (deg f) - MAX < (deg f) - 0 by XREAL_1:10;
A27: ( (f |^ MAX) . X = 0. V & dom (f |^ MAX) = the carrier of V ) by A13, A17, A22, FUNCT_2:def_1, VECTSP11:21;
A28: MAX - MAX < (deg f) - MAX by A24, XREAL_1:9;
A29: (deg f) - MAX = (deg f) -' MAX by A24, XREAL_1:233;
then A30: deg f = ((deg f) -' MAX) + MAX ;
0. V = (f |^ (deg f)) . X by A11, FUNCOP_1:7
.= ((f |^ ((deg f) -' MAX)) * (f |^ MAX)) . X by A30, VECTSP11:20
.= (f |^ ((deg f) -' MAX)) . (0. V) by A27, FUNCT_1:13 ;
then m <= (deg f) -' MAX by A13, A29, A28, A26;
then A31: MAX + m <= MAX + ((deg f) - MAX) by A29, XREAL_1:6;
MAX + m = k + ((I + 1) * m) by A22;
then MAX + m <= MAX + 0 by A21, A31;
hence (f |^ ((deg f) -' m)) . X = 0. V by A13, XREAL_1:6; ::_thesis: verum
end;
end;
end;
hence (f |^ ((deg f) -' m)) . x = 0. V ; ::_thesis: verum
end;
A32: (deg f) - m < (deg f) - 0 by A13, XREAL_1:10;
dom (f |^ ((deg f) -' m)) = [#] V by FUNCT_2:def_1;
then f |^ ((deg f) -' m) = the carrier of V --> (0. V) by A15, FUNCOP_1:11
.= ZeroMap (V,V) by GRCAT_1:def_7 ;
hence contradiction by A14, A32, Def5; ::_thesis: verum
end;
theorem Th17: :: MATRIXJ2:17
for K being Field
for V being VectSp of K
for W being Subspace of V
for f being nilpotent Function of V,V st f | W is Function of W,W holds
f | W is nilpotent Function of W,W
proof
let K be Field; ::_thesis: for V being VectSp of K
for W being Subspace of V
for f being nilpotent Function of V,V st f | W is Function of W,W holds
f | W is nilpotent Function of W,W
let V be VectSp of K; ::_thesis: for W being Subspace of V
for f being nilpotent Function of V,V st f | W is Function of W,W holds
f | W is nilpotent Function of W,W
let W be Subspace of V; ::_thesis: for f being nilpotent Function of V,V st f | W is Function of W,W holds
f | W is nilpotent Function of W,W
let f be nilpotent Function of V,V; ::_thesis: ( f | W is Function of W,W implies f | W is nilpotent Function of W,W )
assume f | W is Function of W,W ; ::_thesis: f | W is nilpotent Function of W,W
then reconsider fW = f | W as Function of W,W ;
consider n being Nat such that
A1: f |^ n = ZeroMap (V,V) by Th13;
[#] W c= [#] V by VECTSP_4:def_2;
then A2: [#] W = ([#] V) /\ ([#] W) by XBOOLE_1:28;
fW |^ n = (ZeroMap (V,V)) | W by A1, VECTSP11:22
.= ( the carrier of V --> (0. V)) | ([#] W) by GRCAT_1:def_7
.= ( the carrier of V /\ ([#] W)) --> (0. V) by FUNCOP_1:12
.= the carrier of W --> (0. W) by A2, VECTSP_4:11
.= ZeroMap (W,W) by GRCAT_1:def_7 ;
hence f | W is nilpotent Function of W,W by Th13; ::_thesis: verum
end;
theorem Th18: :: MATRIXJ2:18
for n being Nat
for K being Field
for V being VectSp of K
for W being Subspace of V
for f being nilpotent linear-transformation of V,V
for fI being nilpotent Function of (im (f |^ n)),(im (f |^ n)) st fI = f | (im (f |^ n)) & n <= deg f holds
(deg fI) + n = deg f
proof
let n be Nat; ::_thesis: for K being Field
for V being VectSp of K
for W being Subspace of V
for f being nilpotent linear-transformation of V,V
for fI being nilpotent Function of (im (f |^ n)),(im (f |^ n)) st fI = f | (im (f |^ n)) & n <= deg f holds
(deg fI) + n = deg f
let K be Field; ::_thesis: for V being VectSp of K
for W being Subspace of V
for f being nilpotent linear-transformation of V,V
for fI being nilpotent Function of (im (f |^ n)),(im (f |^ n)) st fI = f | (im (f |^ n)) & n <= deg f holds
(deg fI) + n = deg f
let V be VectSp of K; ::_thesis: for W being Subspace of V
for f being nilpotent linear-transformation of V,V
for fI being nilpotent Function of (im (f |^ n)),(im (f |^ n)) st fI = f | (im (f |^ n)) & n <= deg f holds
(deg fI) + n = deg f
let W be Subspace of V; ::_thesis: for f being nilpotent linear-transformation of V,V
for fI being nilpotent Function of (im (f |^ n)),(im (f |^ n)) st fI = f | (im (f |^ n)) & n <= deg f holds
(deg fI) + n = deg f
let f be nilpotent linear-transformation of V,V; ::_thesis: for fI being nilpotent Function of (im (f |^ n)),(im (f |^ n)) st fI = f | (im (f |^ n)) & n <= deg f holds
(deg fI) + n = deg f
set IM = im (f |^ n);
let fI be nilpotent Function of (im (f |^ n)),(im (f |^ n)); ::_thesis: ( fI = f | (im (f |^ n)) & n <= deg f implies (deg fI) + n = deg f )
assume A1: fI = f | (im (f |^ n)) ; ::_thesis: ( not n <= deg f or (deg fI) + n = deg f )
set D = deg f;
assume n <= deg f ; ::_thesis: (deg fI) + n = deg f
then reconsider Dn = (deg f) - n as Element of NAT by NAT_1:21;
A2: now__::_thesis:_for_x_being_set_st_x_in_dom_(fI_|^_Dn)_holds_
0._(im_(f_|^_n))_=_(fI_|^_Dn)_._x
let x be set ; ::_thesis: ( x in dom (fI |^ Dn) implies 0. (im (f |^ n)) = (fI |^ Dn) . x )
assume x in dom (fI |^ Dn) ; ::_thesis: 0. (im (f |^ n)) = (fI |^ Dn) . x
then reconsider X = x as Vector of (im (f |^ n)) by FUNCT_2:def_1;
reconsider v = X as Vector of V by VECTSP_4:10;
A3: dom (f |^ n) = the carrier of V by FUNCT_2:def_1;
X in im (f |^ n) by STRUCT_0:def_5;
then consider w being Element of V such that
A4: v = (f |^ n) . w by RANKNULL:13;
(f |^ (deg f)) . w = (ZeroMap (V,V)) . w by Def5
.= ( the carrier of V --> (0. V)) . w by GRCAT_1:def_7
.= 0. V by FUNCOP_1:7 ;
hence 0. (im (f |^ n)) = (f |^ (Dn + n)) . w by VECTSP_4:11
.= ((f |^ Dn) * (f |^ n)) . w by VECTSP11:20
.= (f |^ Dn) . v by A4, A3, FUNCT_1:13
.= ((f |^ Dn) | (im (f |^ n))) . X by FUNCT_1:49
.= (fI |^ Dn) . x by A1, VECTSP11:22 ;
::_thesis: verum
end;
dom (fI |^ Dn) = [#] (im (f |^ n)) by FUNCT_2:def_1;
then fI |^ Dn = the carrier of (im (f |^ n)) --> (0. (im (f |^ n))) by A2, FUNCOP_1:11
.= ZeroMap ((im (f |^ n)),(im (f |^ n))) by GRCAT_1:def_7 ;
then A5: deg fI <= Dn by Def5;
deg fI = Dn
proof
set DI = deg fI;
A6: dom (f |^ n) = the carrier of V by FUNCT_2:def_1;
assume deg fI <> Dn ; ::_thesis: contradiction
then deg fI < Dn by A5, XXREAL_0:1;
then A7: (deg fI) + n < Dn + n by XREAL_1:6;
consider v being Vector of V such that
A8: for i being Nat st i < deg f holds
(f |^ i) . v <> 0. V by Th16;
(f |^ n) . v in im (f |^ n) by RANKNULL:13;
then A9: (f |^ n) . v in the carrier of (im (f |^ n)) by STRUCT_0:def_5;
fI |^ (deg fI) = ZeroMap ((im (f |^ n)),(im (f |^ n))) by Def5
.= the carrier of (im (f |^ n)) --> (0. (im (f |^ n))) by GRCAT_1:def_7 ;
then 0. (im (f |^ n)) = (fI |^ (deg fI)) . ((f |^ n) . v) by A9, FUNCOP_1:7
.= ((f |^ (deg fI)) | (im (f |^ n))) . ((f |^ n) . v) by A1, VECTSP11:22
.= (f |^ (deg fI)) . ((f |^ n) . v) by A9, FUNCT_1:49
.= ((f |^ (deg fI)) * (f |^ n)) . v by A6, FUNCT_1:13
.= (f |^ ((deg fI) + n)) . v by VECTSP11:20 ;
hence contradiction by A7, A8, VECTSP_4:11; ::_thesis: verum
end;
hence (deg fI) + n = deg f ; ::_thesis: verum
end;
theorem Th19: :: MATRIXJ2:19
for i being Nat
for K being Field
for V1, V2 being finite-dimensional VectSp of K
for W1, W2 being Subspace of V1
for U1, U2 being Subspace of V2
for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for bw1 being OrdBasis of W1
for bw2 being OrdBasis of W2
for Bu1 being FinSequence of U1
for Bu2 being FinSequence of U2
for M being Matrix of len b1, len B2,K
for M1 being Matrix of len bw1, len Bu1,K
for M2 being Matrix of len bw2, len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 holds
( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
proof
let i be Nat; ::_thesis: for K being Field
for V1, V2 being finite-dimensional VectSp of K
for W1, W2 being Subspace of V1
for U1, U2 being Subspace of V2
for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for bw1 being OrdBasis of W1
for bw2 being OrdBasis of W2
for Bu1 being FinSequence of U1
for Bu2 being FinSequence of U2
for M being Matrix of len b1, len B2,K
for M1 being Matrix of len bw1, len Bu1,K
for M2 being Matrix of len bw2, len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 holds
( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
let K be Field; ::_thesis: for V1, V2 being finite-dimensional VectSp of K
for W1, W2 being Subspace of V1
for U1, U2 being Subspace of V2
for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for bw1 being OrdBasis of W1
for bw2 being OrdBasis of W2
for Bu1 being FinSequence of U1
for Bu2 being FinSequence of U2
for M being Matrix of len b1, len B2,K
for M1 being Matrix of len bw1, len Bu1,K
for M2 being Matrix of len bw2, len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 holds
( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for W1, W2 being Subspace of V1
for U1, U2 being Subspace of V2
for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for bw1 being OrdBasis of W1
for bw2 being OrdBasis of W2
for Bu1 being FinSequence of U1
for Bu2 being FinSequence of U2
for M being Matrix of len b1, len B2,K
for M1 being Matrix of len bw1, len Bu1,K
for M2 being Matrix of len bw2, len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 holds
( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
let W1, W2 be Subspace of V1; ::_thesis: for U1, U2 being Subspace of V2
for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for bw1 being OrdBasis of W1
for bw2 being OrdBasis of W2
for Bu1 being FinSequence of U1
for Bu2 being FinSequence of U2
for M being Matrix of len b1, len B2,K
for M1 being Matrix of len bw1, len Bu1,K
for M2 being Matrix of len bw2, len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 holds
( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
let U1, U2 be Subspace of V2; ::_thesis: for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for bw1 being OrdBasis of W1
for bw2 being OrdBasis of W2
for Bu1 being FinSequence of U1
for Bu2 being FinSequence of U2
for M being Matrix of len b1, len B2,K
for M1 being Matrix of len bw1, len Bu1,K
for M2 being Matrix of len bw2, len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 holds
( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
let b1 be OrdBasis of V1; ::_thesis: for B2 being FinSequence of V2
for bw1 being OrdBasis of W1
for bw2 being OrdBasis of W2
for Bu1 being FinSequence of U1
for Bu2 being FinSequence of U2
for M being Matrix of len b1, len B2,K
for M1 being Matrix of len bw1, len Bu1,K
for M2 being Matrix of len bw2, len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 holds
( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
let B2 be FinSequence of V2; ::_thesis: for bw1 being OrdBasis of W1
for bw2 being OrdBasis of W2
for Bu1 being FinSequence of U1
for Bu2 being FinSequence of U2
for M being Matrix of len b1, len B2,K
for M1 being Matrix of len bw1, len Bu1,K
for M2 being Matrix of len bw2, len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 holds
( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
let bw1 be OrdBasis of W1; ::_thesis: for bw2 being OrdBasis of W2
for Bu1 being FinSequence of U1
for Bu2 being FinSequence of U2
for M being Matrix of len b1, len B2,K
for M1 being Matrix of len bw1, len Bu1,K
for M2 being Matrix of len bw2, len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 holds
( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
let bw2 be OrdBasis of W2; ::_thesis: for Bu1 being FinSequence of U1
for Bu2 being FinSequence of U2
for M being Matrix of len b1, len B2,K
for M1 being Matrix of len bw1, len Bu1,K
for M2 being Matrix of len bw2, len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 holds
( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
let Bu1 be FinSequence of U1; ::_thesis: for Bu2 being FinSequence of U2
for M being Matrix of len b1, len B2,K
for M1 being Matrix of len bw1, len Bu1,K
for M2 being Matrix of len bw2, len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 holds
( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
let Bu2 be FinSequence of U2; ::_thesis: for M being Matrix of len b1, len B2,K
for M1 being Matrix of len bw1, len Bu1,K
for M2 being Matrix of len bw2, len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 holds
( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
let M be Matrix of len b1, len B2,K; ::_thesis: for M1 being Matrix of len bw1, len Bu1,K
for M2 being Matrix of len bw2, len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 holds
( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
let M1 be Matrix of len bw1, len Bu1,K; ::_thesis: for M2 being Matrix of len bw2, len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 holds
( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
let M2 be Matrix of len bw2, len Bu2,K; ::_thesis: ( b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M = block_diagonal (<*M1,M2*>,(0. K)) & width M1 = len Bu1 & width M2 = len Bu2 implies ( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) ) )
assume that
A1: b1 = bw1 ^ bw2 and
A2: B2 = Bu1 ^ Bu2 and
A3: M = block_diagonal (<*M1,M2*>,(0. K)) and
A4: width M1 = len Bu1 and
A5: width M2 = len Bu2 ; ::_thesis: ( ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) & ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) ) )
A6: dom bw1 c= dom b1 by A1, FINSEQ_1:26;
( rng Bu2 c= the carrier of U2 & the carrier of U2 c= the carrier of V2 ) by FINSEQ_1:def_4, VECTSP_4:def_2;
then A7: rng Bu2 c= the carrier of V2 by XBOOLE_1:1;
( rng Bu1 c= the carrier of U1 & the carrier of U1 c= the carrier of V2 ) by FINSEQ_1:def_4, VECTSP_4:def_2;
then rng Bu1 c= the carrier of V2 by XBOOLE_1:1;
then reconsider bu1 = Bu1, bu2 = Bu2 as FinSequence of V2 by A7, FINSEQ_1:def_4;
set F1 = Mx2Tran (M1,bw1,Bu1);
set F = Mx2Tran (M,b1,B2);
A8: ( dom b1 = Seg (len b1) & len (1. (K,(len b1))) = len b1 ) by FINSEQ_1:def_3, MATRIX_1:24;
A9: dom (1. (K,(len bw1))) = Seg (len (1. (K,(len bw1)))) by FINSEQ_1:def_3;
A10: dom (1. (K,(len b1))) = Seg (len (1. (K,(len b1)))) by FINSEQ_1:def_3;
set BI = (len bw1) + i;
A11: ( dom bw2 = Seg (len bw2) & len (1. (K,(len bw2))) = len bw2 ) by FINSEQ_1:def_3, MATRIX_1:24;
set F2 = Mx2Tran (M2,bw2,Bu2);
A12: width (1. (K,(len b1))) = len b1 by MATRIX_1:24;
A13: ( len (Line (M2,i)) = width M2 & len ((width M1) |-> (0. K)) = width M1 ) by CARD_1:def_7;
A14: ( dom bw1 = Seg (len bw1) & len (1. (K,(len bw1))) = len bw1 ) by FINSEQ_1:def_3, MATRIX_1:24;
A15: len M = len b1 by MATRIX_1:def_2;
A16: len M1 = len bw1 by MATRIX_1:def_2;
then A17: dom M1 = dom bw1 by FINSEQ_3:29;
thus ( i in dom bw1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) ) ::_thesis: ( i in dom bw2 implies (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) )
proof
A18: ( len (Line (M1,i)) = width M1 & len ((width M2) |-> (0. K)) = width M2 ) by CARD_1:def_7;
assume A19: i in dom bw1 ; ::_thesis: (Mx2Tran (M,b1,B2)) . (b1 /. i) = (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i)
then A20: Line (M,i) = (Line (M1,i)) ^ ((width M2) |-> (0. K)) by A3, A17, MATRIXJ1:23;
thus (Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line (((LineVec2Mx ((b1 /. i) |-- b1)) * M),1)),B2)) by MATRLIN2:def_3
.= Sum (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len b1))),i))) * M),1)),B2)) by A6, A19, MATRLIN2:19
.= Sum (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len b1))) * M),i))),1)),B2)) by A6, A8, A15, A10, A19, MATRIX_1:24, MATRLIN2:35
.= Sum (lmlt ((Line ((LineVec2Mx (Line (M,i))),1)),B2)) by A15, MATRIXR2:68
.= Sum (lmlt ((Line (M,i)),B2)) by MATRIX15:25
.= Sum ((lmlt ((Line (M1,i)),bu1)) ^ (lmlt (((width M2) |-> (0. K)),bu2))) by A2, A4, A5, A20, A18, MATRLIN2:9
.= (Sum (lmlt ((Line (M1,i)),bu1))) + (Sum (lmlt (((width M2) |-> (0. K)),bu2))) by RLVECT_1:41
.= (Sum (lmlt ((Line (M1,i)),bu1))) + ((0. K) * (Sum bu2)) by A5, MATRLIN2:11
.= (Sum (lmlt ((Line (M1,i)),bu1))) + (0. V2) by VECTSP_1:14
.= Sum (lmlt ((Line (M1,i)),bu1)) by RLVECT_1:def_4
.= Sum (lmlt ((Line (M1,i)),Bu1)) by MATRLIN2:14, MATRLIN2:15
.= Sum (lmlt ((Line ((LineVec2Mx (Line (M1,i))),1)),Bu1)) by MATRIX15:25
.= Sum (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len bw1))) * M1),i))),1)),Bu1)) by A16, MATRIXR2:68
.= Sum (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len bw1))),i))) * M1),1)),Bu1)) by A14, A9, A16, A19, MATRIX_1:24, MATRLIN2:35
.= Sum (lmlt ((Line (((LineVec2Mx ((bw1 /. i) |-- bw1)) * M1),1)),Bu1)) by A19, MATRLIN2:19
.= (Mx2Tran (M1,bw1,Bu1)) . (bw1 /. i) by MATRLIN2:def_3 ; ::_thesis: verum
end;
A21: dom (1. (K,(len bw2))) = Seg (len (1. (K,(len bw2)))) by FINSEQ_1:def_3;
assume A22: i in dom bw2 ; ::_thesis: (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i)
then A23: (len bw1) + i in dom b1 by A1, FINSEQ_1:28;
A24: len M2 = len bw2 by MATRIX_1:def_2;
then dom M2 = dom bw2 by FINSEQ_3:29;
then A25: Line (M,((len bw1) + i)) = ((width M1) |-> (0. K)) ^ (Line (M2,i)) by A3, A16, A22, MATRIXJ1:23;
thus (Mx2Tran (M,b1,B2)) . (b1 /. (i + (len bw1))) = Sum (lmlt ((Line (((LineVec2Mx ((b1 /. ((len bw1) + i)) |-- b1)) * M),1)),B2)) by MATRLIN2:def_3
.= Sum (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len b1))),((len bw1) + i)))) * M),1)),B2)) by A23, MATRLIN2:19
.= Sum (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len b1))) * M),((len bw1) + i)))),1)),B2)) by A1, A12, A8, A15, A10, A22, FINSEQ_1:28, MATRLIN2:35
.= Sum (lmlt ((Line ((LineVec2Mx (Line (M,((len bw1) + i)))),1)),B2)) by A15, MATRIXR2:68
.= Sum (lmlt ((Line (M,((len bw1) + i))),B2)) by MATRIX15:25
.= Sum ((lmlt (((width M1) |-> (0. K)),bu1)) ^ (lmlt ((Line (M2,i)),bu2))) by A2, A4, A5, A25, A13, MATRLIN2:9
.= (Sum (lmlt (((width M1) |-> (0. K)),bu1))) + (Sum (lmlt ((Line (M2,i)),bu2))) by RLVECT_1:41
.= ((0. K) * (Sum bu1)) + (Sum (lmlt ((Line (M2,i)),bu2))) by A4, MATRLIN2:11
.= (0. V2) + (Sum (lmlt ((Line (M2,i)),bu2))) by VECTSP_1:14
.= Sum (lmlt ((Line (M2,i)),bu2)) by RLVECT_1:def_4
.= Sum (lmlt ((Line (M2,i)),Bu2)) by MATRLIN2:14, MATRLIN2:15
.= Sum (lmlt ((Line ((LineVec2Mx (Line (M2,i))),1)),Bu2)) by MATRIX15:25
.= Sum (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len bw2))) * M2),i))),1)),Bu2)) by A24, MATRIXR2:68
.= Sum (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len bw2))),i))) * M2),1)),Bu2)) by A11, A21, A24, A22, MATRIX_1:24, MATRLIN2:35
.= Sum (lmlt ((Line (((LineVec2Mx ((bw2 /. i) |-- bw2)) * M2),1)),Bu2)) by A22, MATRLIN2:19
.= (Mx2Tran (M2,bw2,Bu2)) . (bw2 /. i) by MATRLIN2:def_3 ; ::_thesis: verum
end;
theorem Th20: :: MATRIXJ2:20
for K being Field
for V1, V2 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for M being Matrix of len b1, len B2,K
for F being FinSequence_of_Matrix of K st M = block_diagonal (F,(0. K)) holds
for i, m being Nat st i in dom b1 & m = min ((Len F),i) holds
( (Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))))) & len ((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))) = width (F . m) )
proof
let K be Field; ::_thesis: for V1, V2 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for M being Matrix of len b1, len B2,K
for F being FinSequence_of_Matrix of K st M = block_diagonal (F,(0. K)) holds
for i, m being Nat st i in dom b1 & m = min ((Len F),i) holds
( (Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))))) & len ((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))) = width (F . m) )
let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for M being Matrix of len b1, len B2,K
for F being FinSequence_of_Matrix of K st M = block_diagonal (F,(0. K)) holds
for i, m being Nat st i in dom b1 & m = min ((Len F),i) holds
( (Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))))) & len ((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))) = width (F . m) )
let b1 be OrdBasis of V1; ::_thesis: for B2 being FinSequence of V2
for M being Matrix of len b1, len B2,K
for F being FinSequence_of_Matrix of K st M = block_diagonal (F,(0. K)) holds
for i, m being Nat st i in dom b1 & m = min ((Len F),i) holds
( (Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))))) & len ((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))) = width (F . m) )
let B2 be FinSequence of V2; ::_thesis: for M being Matrix of len b1, len B2,K
for F being FinSequence_of_Matrix of K st M = block_diagonal (F,(0. K)) holds
for i, m being Nat st i in dom b1 & m = min ((Len F),i) holds
( (Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))))) & len ((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))) = width (F . m) )
set ONE = 1. (K,(len b1));
let M be Matrix of len b1, len B2,K; ::_thesis: for F being FinSequence_of_Matrix of K st M = block_diagonal (F,(0. K)) holds
for i, m being Nat st i in dom b1 & m = min ((Len F),i) holds
( (Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))))) & len ((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))) = width (F . m) )
let F be FinSequence_of_Matrix of K; ::_thesis: ( M = block_diagonal (F,(0. K)) implies for i, m being Nat st i in dom b1 & m = min ((Len F),i) holds
( (Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))))) & len ((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))) = width (F . m) ) )
assume A1: M = block_diagonal (F,(0. K)) ; ::_thesis: for i, m being Nat st i in dom b1 & m = min ((Len F),i) holds
( (Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))))) & len ((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))) = width (F . m) )
A2: width M = Sum (Width F) by A1, MATRIXJ1:def_5;
len (1. (K,(len b1))) = len b1 by MATRIX_1:def_2;
then A3: dom (1. (K,(len b1))) = dom b1 by FINSEQ_3:29;
set L = Len F;
set W = Width F;
let i, m be Nat; ::_thesis: ( i in dom b1 & m = min ((Len F),i) implies ( (Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))))) & len ((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))) = width (F . m) ) )
assume that
A4: i in dom b1 and
A5: m = min ((Len F),i) ; ::_thesis: ( (Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))))) & len ((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))) = width (F . m) )
A6: ( 1 <= i & i <= len b1 ) by A4, FINSEQ_3:25;
then A7: len M = len b1 by MATRIX_1:23;
set Fm = F . m;
set Wm1 = Sum ((Width F) | (m -' 1));
A8: ( len ((Sum ((Width F) | (m -' 1))) |-> (0. K)) = Sum ((Width F) | (m -' 1)) & len (Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))) = width (F . m) ) by CARD_1:def_7;
set Wm = Sum ((Width F) | m);
Width F = ((Width F) | m) ^ ((Width F) /^ m) by RFINSEQ:8;
then A9: Sum (Width F) = (Sum ((Width F) | m)) + (Sum ((Width F) /^ m)) by RVSUM_1:75;
then A10: (Sum (Width F)) -' (Sum ((Width F) | m)) = (Sum (Width F)) - (Sum ((Width F) | m)) by NAT_1:11, XREAL_1:233
.= Sum ((Width F) /^ m) by A9 ;
then A11: len (((Sum (Width F)) -' (Sum ((Width F) | m))) |-> (0. K)) = Sum ((Width F) /^ m) by CARD_1:def_7;
A12: ( dom b1 = Seg (len b1) & len M = Sum (Len F) ) by A1, FINSEQ_1:def_3, MATRIXJ1:def_5;
then A13: m in dom (Len F) by A4, A5, A7, MATRIXJ1:def_1;
then A14: 1 <= m by FINSEQ_3:25;
( dom (Len F) = dom F & dom (Width F) = dom F ) by MATRIXJ1:def_3, MATRIXJ1:def_4;
then ( m <= len (Width F) & (Width F) . m = width (F . m) ) by A13, FINSEQ_3:25, MATRIXJ1:def_4;
then Width F = (((Width F) | (m -' 1)) ^ <*(width (F . m))*>) ^ ((Width F) /^ m) by A5, A14, POLYNOM4:1;
then A15: Sum (Width F) = (Sum (((Width F) | (m -' 1)) ^ <*(width (F . m))*>)) + (Sum ((Width F) /^ m)) by RVSUM_1:75
.= ((Sum ((Width F) | (m -' 1))) + (width (F . m))) + (Sum ((Width F) /^ m)) by RVSUM_1:74 ;
then A16: len (((Sum ((Width F) | (m -' 1))) |-> (0. K)) ^ (Line ((F . m),(i -' (Sum (Len (F | (m -' 1)))))))) = Sum ((Width F) | m) by A9, CARD_1:def_7;
set B2W = B2 | (Sum ((Width F) | m));
A17: (B2 | (Sum ((Width F) | m))) /^ (Sum ((Width F) | (m -' 1))) = (B2 | (Sum (Width (F | m)))) /^ (Sum ((Width F) | (m -' 1))) by MATRIXJ1:21
.= (B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1)))) by MATRIXJ1:21 ;
A18: width M = len B2 by A6, MATRIX_1:23;
then A19: len (B2 | (Sum ((Width F) | m))) = Sum ((Width F) | m) by A2, A9, FINSEQ_1:59, NAT_1:11;
Sum (Width F) = (Sum ((Width F) | (m -' 1))) + ((width (F . m)) + (Sum ((Width F) /^ m))) by A15;
then A20: len (B2 | (Sum ((Width F) | (m -' 1)))) = Sum ((Width F) | (m -' 1)) by A2, A18, FINSEQ_1:59, NAT_1:11;
Sum ((Width F) | m) >= Sum ((Width F) | (m -' 1)) by A9, A15, NAT_1:11;
then Seg (Sum ((Width F) | (m -' 1))) c= Seg (Sum ((Width F) | m)) by FINSEQ_1:5;
then (B2 | (Sum ((Width F) | m))) | (Sum ((Width F) | (m -' 1))) = B2 | (Sum ((Width F) | (m -' 1))) by RELAT_1:74;
then A21: B2 | (Sum ((Width F) | m)) = (B2 | (Sum ((Width F) | (m -' 1)))) ^ ((B2 | (Sum ((Width F) | m))) /^ (Sum ((Width F) | (m -' 1)))) by RFINSEQ:8;
then A22: len (B2 | (Sum ((Width F) | m))) = (len (B2 | (Sum ((Width F) | (m -' 1))))) + (len ((B2 | (Sum ((Width F) | m))) /^ (Sum ((Width F) | (m -' 1))))) by FINSEQ_1:22;
A23: B2 = (B2 | (Sum ((Width F) | m))) ^ (B2 /^ (Sum ((Width F) | m))) by RFINSEQ:8;
then A24: len B2 = (len (B2 | (Sum ((Width F) | m)))) + (len (B2 /^ (Sum ((Width F) | m)))) by FINSEQ_1:22;
(Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line (((LineVec2Mx ((b1 /. i) |-- b1)) * M),1)),B2)) by MATRLIN2:def_3
.= Sum (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len b1))),i))) * M),1)),B2)) by A4, MATRLIN2:19
.= Sum (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len b1))) * M),i))),1)),B2)) by A4, A6, A3, A7, MATRIX_1:23, MATRLIN2:35
.= Sum (lmlt ((Line ((LineVec2Mx (Line (M,i))),1)),B2)) by A7, MATRIXR2:68
.= Sum (lmlt ((Line (M,i)),B2)) by MATRIX15:25
.= Sum (lmlt (((((Sum (Width (F | (m -' 1)))) |-> (0. K)) ^ (Line ((F . m),(i -' (Sum (Len (F | (m -' 1)))))))) ^ (((Sum (Width F)) -' (Sum (Width (F | m)))) |-> (0. K))),B2)) by A1, A4, A5, A7, A12, MATRIXJ1:37
.= Sum (lmlt (((((Sum ((Width F) | (m -' 1))) |-> (0. K)) ^ (Line ((F . m),(i -' (Sum (Len (F | (m -' 1)))))))) ^ (((Sum (Width F)) -' (Sum (Width (F | m)))) |-> (0. K))),B2)) by MATRIXJ1:21
.= Sum (lmlt (((((Sum ((Width F) | (m -' 1))) |-> (0. K)) ^ (Line ((F . m),(i -' (Sum (Len (F | (m -' 1)))))))) ^ (((Sum (Width F)) -' (Sum ((Width F) | m))) |-> (0. K))),B2)) by MATRIXJ1:21
.= Sum ((lmlt ((((Sum ((Width F) | (m -' 1))) |-> (0. K)) ^ (Line ((F . m),(i -' (Sum (Len (F | (m -' 1)))))))),(B2 | (Sum ((Width F) | m))))) ^ (lmlt ((((Sum (Width F)) -' (Sum ((Width F) | m))) |-> (0. K)),(B2 /^ (Sum ((Width F) | m)))))) by A2, A18, A9, A23, A24, A19, A11, A16, MATRLIN2:9
.= (Sum (lmlt ((((Sum ((Width F) | (m -' 1))) |-> (0. K)) ^ (Line ((F . m),(i -' (Sum (Len (F | (m -' 1)))))))),(B2 | (Sum ((Width F) | m)))))) + (Sum (lmlt ((((Sum (Width F)) -' (Sum ((Width F) | m))) |-> (0. K)),(B2 /^ (Sum ((Width F) | m)))))) by RLVECT_1:41
.= (Sum (lmlt ((((Sum ((Width F) | (m -' 1))) |-> (0. K)) ^ (Line ((F . m),(i -' (Sum (Len (F | (m -' 1)))))))),(B2 | (Sum ((Width F) | m)))))) + ((0. K) * (Sum (B2 /^ (Sum ((Width F) | m))))) by A2, A18, A9, A10, A24, A19, MATRLIN2:11
.= (Sum (lmlt ((((Sum ((Width F) | (m -' 1))) |-> (0. K)) ^ (Line ((F . m),(i -' (Sum (Len (F | (m -' 1)))))))),(B2 | (Sum ((Width F) | m)))))) + (0. V2) by VECTSP_1:14
.= Sum (lmlt ((((Sum ((Width F) | (m -' 1))) |-> (0. K)) ^ (Line ((F . m),(i -' (Sum (Len (F | (m -' 1)))))))),(B2 | (Sum ((Width F) | m))))) by RLVECT_1:def_4
.= Sum ((lmlt (((Sum ((Width F) | (m -' 1))) |-> (0. K)),(B2 | (Sum ((Width F) | (m -' 1)))))) ^ (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum ((Width F) | m))) /^ (Sum ((Width F) | (m -' 1))))))) by A9, A15, A19, A21, A22, A20, A8, MATRLIN2:9
.= (Sum (lmlt (((Sum ((Width F) | (m -' 1))) |-> (0. K)),(B2 | (Sum ((Width F) | (m -' 1))))))) + (Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum ((Width F) | m))) /^ (Sum ((Width F) | (m -' 1))))))) by RLVECT_1:41
.= ((0. K) * (Sum (B2 | (Sum ((Width F) | (m -' 1)))))) + (Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum ((Width F) | m))) /^ (Sum ((Width F) | (m -' 1))))))) by A20, MATRLIN2:11
.= (0. V2) + (Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum ((Width F) | m))) /^ (Sum ((Width F) | (m -' 1))))))) by VECTSP_1:14
.= Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum ((Width F) | m))) /^ (Sum ((Width F) | (m -' 1)))))) by RLVECT_1:def_4 ;
hence ( (Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line ((F . m),(i -' (Sum (Len (F | (m -' 1))))))),((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))))) & len ((B2 | (Sum (Width (F | m)))) /^ (Sum (Width (F | (m -' 1))))) = width (F . m) ) by A9, A15, A19, A22, A20, A17; ::_thesis: verum
end;
theorem Th21: :: MATRIXJ2:21
for K being Field
for L being Element of K
for V1 being finite-dimensional VectSp of K
for B1 being FinSequence of V1 st len B1 in dom B1 holds
Sum (lmlt ((Line ((Jordan_block (L,(len B1))),(len B1))),B1)) = L * (B1 /. (len B1))
proof
let K be Field; ::_thesis: for L being Element of K
for V1 being finite-dimensional VectSp of K
for B1 being FinSequence of V1 st len B1 in dom B1 holds
Sum (lmlt ((Line ((Jordan_block (L,(len B1))),(len B1))),B1)) = L * (B1 /. (len B1))
let L be Element of K; ::_thesis: for V1 being finite-dimensional VectSp of K
for B1 being FinSequence of V1 st len B1 in dom B1 holds
Sum (lmlt ((Line ((Jordan_block (L,(len B1))),(len B1))),B1)) = L * (B1 /. (len B1))
let V1 be finite-dimensional VectSp of K; ::_thesis: for B1 being FinSequence of V1 st len B1 in dom B1 holds
Sum (lmlt ((Line ((Jordan_block (L,(len B1))),(len B1))),B1)) = L * (B1 /. (len B1))
let B1 be FinSequence of V1; ::_thesis: ( len B1 in dom B1 implies Sum (lmlt ((Line ((Jordan_block (L,(len B1))),(len B1))),B1)) = L * (B1 /. (len B1)) )
set N = len B1;
assume A1: len B1 in dom B1 ; ::_thesis: Sum (lmlt ((Line ((Jordan_block (L,(len B1))),(len B1))),B1)) = L * (B1 /. (len B1))
set J = Jordan_block (L,(len B1));
set ONE = 1. (K,(len B1));
thus Sum (lmlt ((Line ((Jordan_block (L,(len B1))),(len B1))),B1)) = Sum (lmlt ((L * (Line ((1. (K,(len B1))),(len B1)))),B1)) by Th5
.= L * (Sum (lmlt ((Line ((1. (K,(len B1))),(len B1))),B1))) by MATRLIN2:13
.= L * (B1 /. (len B1)) by A1, MATRLIN2:16 ; ::_thesis: verum
end;
theorem Th22: :: MATRIXJ2:22
for i being Nat
for K being Field
for L being Element of K
for V1 being finite-dimensional VectSp of K
for B1 being FinSequence of V1 st i in dom B1 & i <> len B1 holds
Sum (lmlt ((Line ((Jordan_block (L,(len B1))),i)),B1)) = (L * (B1 /. i)) + (B1 /. (i + 1))
proof
let i be Nat; ::_thesis: for K being Field
for L being Element of K
for V1 being finite-dimensional VectSp of K
for B1 being FinSequence of V1 st i in dom B1 & i <> len B1 holds
Sum (lmlt ((Line ((Jordan_block (L,(len B1))),i)),B1)) = (L * (B1 /. i)) + (B1 /. (i + 1))
let K be Field; ::_thesis: for L being Element of K
for V1 being finite-dimensional VectSp of K
for B1 being FinSequence of V1 st i in dom B1 & i <> len B1 holds
Sum (lmlt ((Line ((Jordan_block (L,(len B1))),i)),B1)) = (L * (B1 /. i)) + (B1 /. (i + 1))
let L be Element of K; ::_thesis: for V1 being finite-dimensional VectSp of K
for B1 being FinSequence of V1 st i in dom B1 & i <> len B1 holds
Sum (lmlt ((Line ((Jordan_block (L,(len B1))),i)),B1)) = (L * (B1 /. i)) + (B1 /. (i + 1))
let V1 be finite-dimensional VectSp of K; ::_thesis: for B1 being FinSequence of V1 st i in dom B1 & i <> len B1 holds
Sum (lmlt ((Line ((Jordan_block (L,(len B1))),i)),B1)) = (L * (B1 /. i)) + (B1 /. (i + 1))
let B1 be FinSequence of V1; ::_thesis: ( i in dom B1 & i <> len B1 implies Sum (lmlt ((Line ((Jordan_block (L,(len B1))),i)),B1)) = (L * (B1 /. i)) + (B1 /. (i + 1)) )
assume that
A1: i in dom B1 and
A2: i <> len B1 ; ::_thesis: Sum (lmlt ((Line ((Jordan_block (L,(len B1))),i)),B1)) = (L * (B1 /. i)) + (B1 /. (i + 1))
set N = len B1;
A3: dom B1 = Seg (len B1) by FINSEQ_1:def_3;
i <= len B1 by A1, FINSEQ_3:25;
then i < len B1 by A2, XXREAL_0:1;
then ( 1 <= i + 1 & i + 1 <= len B1 ) by NAT_1:11, NAT_1:13;
then A4: i + 1 in dom B1 by A3;
set ONE = 1. (K,(len B1));
A5: len (Line ((1. (K,(len B1))),(i + 1))) = width (1. (K,(len B1))) by CARD_1:def_7;
width (1. (K,(len B1))) = len B1 by MATRIX_1:24;
then A6: dom (Line ((1. (K,(len B1))),(i + 1))) = dom B1 by A5, FINSEQ_3:29;
( len (L * (Line ((1. (K,(len B1))),i))) = len (Line ((1. (K,(len B1))),i)) & len (Line ((1. (K,(len B1))),i)) = width (1. (K,(len B1))) ) by CARD_1:def_7, MATRIXR1:16;
then dom (L * (Line ((1. (K,(len B1))),i))) = dom (Line ((1. (K,(len B1))),(i + 1))) by A5, FINSEQ_3:29;
then A7: dom (lmlt ((L * (Line ((1. (K,(len B1))),i))),B1)) = dom B1 by A6, MATRLIN:12;
dom (lmlt ((Line ((1. (K,(len B1))),(i + 1))),B1)) = dom B1 by A6, MATRLIN:12;
then A8: len (lmlt ((L * (Line ((1. (K,(len B1))),i))),B1)) = len (lmlt ((Line ((1. (K,(len B1))),(i + 1))),B1)) by A7, FINSEQ_3:29;
thus Sum (lmlt ((Line ((Jordan_block (L,(len B1))),i)),B1)) = Sum (lmlt (((L * (Line ((1. (K,(len B1))),i))) + (Line ((1. (K,(len B1))),(i + 1)))),B1)) by A1, A2, A3, Th4
.= Sum ((lmlt ((L * (Line ((1. (K,(len B1))),i))),B1)) + (lmlt ((Line ((1. (K,(len B1))),(i + 1))),B1))) by MATRLIN2:7
.= (Sum (lmlt ((L * (Line ((1. (K,(len B1))),i))),B1))) + (Sum (lmlt ((Line ((1. (K,(len B1))),(i + 1))),B1))) by A8, MATRLIN2:10
.= (L * (Sum (lmlt ((Line ((1. (K,(len B1))),i)),B1)))) + (Sum (lmlt ((Line ((1. (K,(len B1))),(i + 1))),B1))) by MATRLIN2:13
.= (L * (B1 /. i)) + (Sum (lmlt ((Line ((1. (K,(len B1))),(i + 1))),B1))) by A1, MATRLIN2:16
.= (L * (B1 /. i)) + (B1 /. (i + 1)) by A4, MATRLIN2:16 ; ::_thesis: verum
end;
theorem Th23: :: MATRIXJ2:23
for n being Nat
for K being Field
for L being Element of K
for V1, V2 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for M being Matrix of len b1, len B2,K st M = Jordan_block (L,n) holds
for i being Nat st i in dom b1 holds
( ( i = len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
proof
let n be Nat; ::_thesis: for K being Field
for L being Element of K
for V1, V2 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for M being Matrix of len b1, len B2,K st M = Jordan_block (L,n) holds
for i being Nat st i in dom b1 holds
( ( i = len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
let K be Field; ::_thesis: for L being Element of K
for V1, V2 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for M being Matrix of len b1, len B2,K st M = Jordan_block (L,n) holds
for i being Nat st i in dom b1 holds
( ( i = len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
let L be Element of K; ::_thesis: for V1, V2 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for M being Matrix of len b1, len B2,K st M = Jordan_block (L,n) holds
for i being Nat st i in dom b1 holds
( ( i = len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for M being Matrix of len b1, len B2,K st M = Jordan_block (L,n) holds
for i being Nat st i in dom b1 holds
( ( i = len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
let b1 be OrdBasis of V1; ::_thesis: for B2 being FinSequence of V2
for M being Matrix of len b1, len B2,K st M = Jordan_block (L,n) holds
for i being Nat st i in dom b1 holds
( ( i = len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
let B2 be FinSequence of V2; ::_thesis: for M being Matrix of len b1, len B2,K st M = Jordan_block (L,n) holds
for i being Nat st i in dom b1 holds
( ( i = len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
set ONE = 1. (K,(len b1));
set J = Jordan_block (L,n);
let M be Matrix of len b1, len B2,K; ::_thesis: ( M = Jordan_block (L,n) implies for i being Nat st i in dom b1 holds
( ( i = len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) ) )
assume A1: M = Jordan_block (L,n) ; ::_thesis: for i being Nat st i in dom b1 holds
( ( i = len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
A2: len M = n by A1, MATRIX_1:24;
len (1. (K,(len b1))) = len b1 by MATRIX_1:def_2;
then A3: dom (1. (K,(len b1))) = dom b1 by FINSEQ_3:29;
let i be Nat; ::_thesis: ( i in dom b1 implies ( ( i = len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) ) )
assume A4: i in dom b1 ; ::_thesis: ( ( i = len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
A5: len M = len b1 by MATRIX_1:25;
A6: (Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line (((LineVec2Mx ((b1 /. i) |-- b1)) * M),1)),B2)) by MATRLIN2:def_3
.= Sum (lmlt ((Line (((LineVec2Mx (Line ((1. (K,(len b1))),i))) * M),1)),B2)) by A4, MATRLIN2:19
.= Sum (lmlt ((Line ((LineVec2Mx (Line (((1. (K,(len b1))) * M),i))),1)),B2)) by A4, A3, A5, MATRIX_1:24, MATRLIN2:35
.= Sum (lmlt ((Line ((LineVec2Mx (Line (M,i))),1)),B2)) by A5, MATRIXR2:68
.= Sum (lmlt ((Line (M,i)),B2)) by MATRIX15:25 ;
dom b1 = Seg (len b1) by FINSEQ_1:def_3;
then n <> 0 by A4, A5, A2;
then A7: ( width (Jordan_block (L,n)) = n & width (Jordan_block (L,n)) = len B2 ) by A1, A5, A2, MATRIX_1:20;
then dom B2 = dom b1 by A5, A2, FINSEQ_3:29;
hence ( ( i = len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> len b1 implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) ) by A1, A4, A5, A2, A7, A6, Th21, Th22; ::_thesis: verum
end;
theorem Th24: :: MATRIXJ2:24
for K being Field
for L being Element of K
for V1, V2 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len B2,K st M = block_diagonal (J,(0. K)) holds
for i, m being Nat st i in dom b1 & m = min ((Len J),i) holds
( ( i = Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
proof
let K be Field; ::_thesis: for L being Element of K
for V1, V2 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len B2,K st M = block_diagonal (J,(0. K)) holds
for i, m being Nat st i in dom b1 & m = min ((Len J),i) holds
( ( i = Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
let L be Element of K; ::_thesis: for V1, V2 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len B2,K st M = block_diagonal (J,(0. K)) holds
for i, m being Nat st i in dom b1 & m = min ((Len J),i) holds
( ( i = Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1
for B2 being FinSequence of V2
for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len B2,K st M = block_diagonal (J,(0. K)) holds
for i, m being Nat st i in dom b1 & m = min ((Len J),i) holds
( ( i = Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
let b1 be OrdBasis of V1; ::_thesis: for B2 being FinSequence of V2
for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len B2,K st M = block_diagonal (J,(0. K)) holds
for i, m being Nat st i in dom b1 & m = min ((Len J),i) holds
( ( i = Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
let B2 be FinSequence of V2; ::_thesis: for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len B2,K st M = block_diagonal (J,(0. K)) holds
for i, m being Nat st i in dom b1 & m = min ((Len J),i) holds
( ( i = Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
let J be FinSequence_of_Jordan_block of L,K; ::_thesis: for M being Matrix of len b1, len B2,K st M = block_diagonal (J,(0. K)) holds
for i, m being Nat st i in dom b1 & m = min ((Len J),i) holds
( ( i = Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
let M be Matrix of len b1, len B2,K; ::_thesis: ( M = block_diagonal (J,(0. K)) implies for i, m being Nat st i in dom b1 & m = min ((Len J),i) holds
( ( i = Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) ) )
assume A1: M = block_diagonal (J,(0. K)) ; ::_thesis: for i, m being Nat st i in dom b1 & m = min ((Len J),i) holds
( ( i = Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
A2: ( dom b1 = Seg (len b1) & len M = Sum (Len J) ) by A1, FINSEQ_1:def_3, MATRIXJ1:def_5;
let i, m be Nat; ::_thesis: ( i in dom b1 & m = min ((Len J),i) implies ( ( i = Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) ) )
assume that
A3: i in dom b1 and
A4: m = min ((Len J),i) ; ::_thesis: ( ( i = Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) & ( i <> Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) ) )
set Sm = Sum (Len (J | m));
A5: 1 <= i by A3, FINSEQ_3:25;
A6: i <= len b1 by A3, FINSEQ_3:25;
then A7: len M = len b1 by A5, MATRIX_1:23;
then A8: m in dom (Len J) by A3, A4, A2, MATRIXJ1:def_1;
then A9: (Len J) . m = len (J . m) by MATRIXJ1:def_3;
set S = Sum (Len (J | (m -' 1)));
set iS = i -' (Sum (Len (J | (m -' 1))));
set BBB = (B2 | (Sum (Width (J | m)))) /^ (Sum (Width (J | (m -' 1))));
A10: (Mx2Tran (M,b1,B2)) . (b1 /. i) = Sum (lmlt ((Line ((J . m),(i -' (Sum (Len (J | (m -' 1))))))),((B2 | (Sum (Width (J | m)))) /^ (Sum (Width (J | (m -' 1))))))) by A1, A3, A4, Th20;
A11: width M = Sum (Width J) by A1, MATRIXJ1:def_5;
A12: len ((B2 | (Sum (Width (J | m)))) /^ (Sum (Width (J | (m -' 1))))) = width (J . m) by A1, A3, A4, Th20;
A13: (Len J) | m = Len (J | m) by MATRIXJ1:17;
then A14: i <= Sum (Len (J | m)) by A3, A4, A7, A2, MATRIXJ1:def_1;
A15: (Len J) | (m -' 1) = Len (J | (m -' 1)) by MATRIXJ1:17;
then Sum (Len (J | (m -' 1))) < i by A3, A4, A7, A2, MATRIXJ1:7;
then A16: i -' (Sum (Len (J | (m -' 1)))) = i - (Sum (Len (J | (m -' 1)))) by XREAL_1:233;
A17: m -' 1 = m - 1 by A3, A4, A7, A2, MATRIXJ1:7;
then (Len J) | ((m -' 1) + 1) = (Len (J | (m -' 1))) ^ <*((Len J) . m)*> by A15, A8, FINSEQ_5:10;
then A18: Sum (Len (J | m)) = (Sum (Len (J | (m -' 1)))) + (len (J . m)) by A13, A17, A9, RVSUM_1:74;
then (Sum (Len (J | (m -' 1)))) + (i -' (Sum (Len (J | (m -' 1))))) <= (Sum (Len (J | (m -' 1)))) + (len (J . m)) by A3, A4, A13, A7, A2, A16, MATRIXJ1:def_1;
then A19: i -' (Sum (Len (J | (m -' 1)))) <= len (J . m) by XREAL_1:6;
dom (Len J) = dom J by MATRIXJ1:def_3;
then consider n being Nat such that
A20: J . m = Jordan_block (L,n) by A8, Def3;
( m in NAT & m <= len (Len J) ) by A8, FINSEQ_3:25;
then Sum (Len (J | m)) <= Sum ((Len J) | (len (Len J))) by A13, POLYNOM3:18;
then A21: Sum (Len (J | m)) <= Sum (Len J) by FINSEQ_1:58;
A22: Width J = Len J by MATRIXJ1:46;
then A23: (Len J) . m = width (J . m) by A8, MATRIXJ1:def_4;
width M = len B2 by A5, A6, MATRIX_1:23;
then A24: len (B2 | (Sum (Len (J | m)))) = Sum (Len (J | m)) by A22, A11, A21, FINSEQ_1:59;
then A25: i in dom (B2 | (Sum (Len (J | m)))) by A5, A14, FINSEQ_3:25;
A26: len (J . m) = n by A20, MATRIX_1:24;
i -' (Sum (Len (J | (m -' 1)))) <> 0 by A3, A4, A15, A7, A2, A16, MATRIXJ1:7;
then 1 <= i -' (Sum (Len (J | (m -' 1)))) by NAT_1:14;
then A27: i -' (Sum (Len (J | (m -' 1)))) in dom ((B2 | (Sum (Width (J | m)))) /^ (Sum (Width (J | (m -' 1))))) by A12, A9, A23, A19, FINSEQ_3:25;
then A28: ((B2 | (Sum (Width (J | m)))) /^ (Sum (Width (J | (m -' 1))))) /. (i -' (Sum (Len (J | (m -' 1))))) = (B2 | (Sum (Width (J | m)))) /. ((Sum (Width (J | (m -' 1)))) + (i -' (Sum (Len (J | (m -' 1)))))) by FINSEQ_5:27
.= (B2 | (Sum (Width (J | m)))) /. ((Sum (Len (J | (m -' 1)))) + (i -' (Sum (Len (J | (m -' 1)))))) by MATRIXJ1:46
.= (B2 | (Sum ((Len J) | m))) /. i by A22, A16, MATRIXJ1:21
.= B2 /. i by A13, A25, FINSEQ_4:70 ;
hence ( i = Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = L * (B2 /. i) ) by A10, A12, A9, A16, A23, A18, A27, A20, A26, Th21; ::_thesis: ( i <> Sum (Len (J | m)) implies (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) )
assume A29: i <> Sum (Len (J | m)) ; ::_thesis: (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1))
then i < Sum (Len (J | m)) by A14, XXREAL_0:1;
then ( 1 <= i + 1 & i + 1 <= Sum (Len (J | m)) ) by NAT_1:11, NAT_1:13;
then A30: i + 1 in dom (B2 | (Sum (Len (J | m)))) by A24, FINSEQ_3:25;
A31: i -' (Sum (Len (J | (m -' 1)))) <> len (J . m) by A16, A18, A29;
then i -' (Sum (Len (J | (m -' 1)))) < len (J . m) by A19, XXREAL_0:1;
then ( 1 <= (i -' (Sum (Len (J | (m -' 1))))) + 1 & (i -' (Sum (Len (J | (m -' 1))))) + 1 <= len (J . m) ) by NAT_1:11, NAT_1:13;
then (i -' (Sum (Len (J | (m -' 1))))) + 1 in dom ((B2 | (Sum (Width (J | m)))) /^ (Sum (Width (J | (m -' 1))))) by A12, A9, A23, FINSEQ_3:25;
then ((B2 | (Sum (Width (J | m)))) /^ (Sum (Width (J | (m -' 1))))) /. ((i -' (Sum (Len (J | (m -' 1))))) + 1) = (B2 | (Sum (Width (J | m)))) /. ((Sum (Width (J | (m -' 1)))) + ((i -' (Sum (Len (J | (m -' 1))))) + 1)) by FINSEQ_5:27
.= (B2 | (Sum (Width (J | m)))) /. ((Sum ((Len J) | (m -' 1))) + ((i -' (Sum (Len (J | (m -' 1))))) + 1)) by A22, MATRIXJ1:21
.= (B2 | (Sum ((Len J) | m))) /. (i + 1) by A15, A22, A16, MATRIXJ1:21
.= B2 /. (i + 1) by A13, A30, FINSEQ_4:70 ;
hence (Mx2Tran (M,b1,B2)) . (b1 /. i) = (L * (B2 /. i)) + (B2 /. (i + 1)) by A10, A12, A9, A23, A27, A20, A26, A28, A31, Th22; ::_thesis: verum
end;
theorem Th25: :: MATRIXJ2:25
for K being Field
for V1 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of 0. K,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) holds
for m being Nat st ( for i being Nat st i in dom J holds
len (J . i) <= m ) holds
(Mx2Tran (M,b1,b1)) |^ m = ZeroMap (V1,V1)
proof
let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of 0. K,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) holds
for m being Nat st ( for i being Nat st i in dom J holds
len (J . i) <= m ) holds
(Mx2Tran (M,b1,b1)) |^ m = ZeroMap (V1,V1)
let V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of 0. K,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) holds
for m being Nat st ( for i being Nat st i in dom J holds
len (J . i) <= m ) holds
(Mx2Tran (M,b1,b1)) |^ m = ZeroMap (V1,V1)
let b1 be OrdBasis of V1; ::_thesis: for J being FinSequence_of_Jordan_block of 0. K,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) holds
for m being Nat st ( for i being Nat st i in dom J holds
len (J . i) <= m ) holds
(Mx2Tran (M,b1,b1)) |^ m = ZeroMap (V1,V1)
let J be FinSequence_of_Jordan_block of 0. K,K; ::_thesis: for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) holds
for m being Nat st ( for i being Nat st i in dom J holds
len (J . i) <= m ) holds
(Mx2Tran (M,b1,b1)) |^ m = ZeroMap (V1,V1)
let M be Matrix of len b1, len b1,K; ::_thesis: ( M = block_diagonal (J,(0. K)) implies for m being Nat st ( for i being Nat st i in dom J holds
len (J . i) <= m ) holds
(Mx2Tran (M,b1,b1)) |^ m = ZeroMap (V1,V1) )
assume A1: M = block_diagonal (J,(0. K)) ; ::_thesis: for m being Nat st ( for i being Nat st i in dom J holds
len (J . i) <= m ) holds
(Mx2Tran (M,b1,b1)) |^ m = ZeroMap (V1,V1)
reconsider Z = ZeroMap (V1,V1) as linear-transformation of V1,V1 ;
set MT = Mx2Tran (M,b1,b1);
let m be Nat; ::_thesis: ( ( for i being Nat st i in dom J holds
len (J . i) <= m ) implies (Mx2Tran (M,b1,b1)) |^ m = ZeroMap (V1,V1) )
assume A2: for i being Nat st i in dom J holds
len (J . i) <= m ; ::_thesis: (Mx2Tran (M,b1,b1)) |^ m = ZeroMap (V1,V1)
A3: ( dom Z = the carrier of V1 & rng b1 c= the carrier of V1 ) by FUNCT_2:def_1, RELAT_1:def_19;
set MTm = (Mx2Tran (M,b1,b1)) |^ m;
A4: dom ((Mx2Tran (M,b1,b1)) |^ m) = the carrier of V1 by FUNCT_2:def_1;
percases ( len b1 = 0 or len b1 > 0 ) ;
suppose len b1 = 0 ; ::_thesis: (Mx2Tran (M,b1,b1)) |^ m = ZeroMap (V1,V1)
then dim V1 = 0 by MATRLIN2:21;
then (Omega). V1 = (0). V1 by VECTSP_9:29;
then A5: the carrier of V1 = {(0. V1)} by VECTSP_4:def_3;
rng ((Mx2Tran (M,b1,b1)) |^ m) c= the carrier of V1 by RELAT_1:def_19;
then rng ((Mx2Tran (M,b1,b1)) |^ m) = {(0. V1)} by A5, ZFMISC_1:33;
then (Mx2Tran (M,b1,b1)) |^ m = the carrier of V1 --> (0. V1) by A4, FUNCOP_1:9
.= Z by GRCAT_1:def_7 ;
hence (Mx2Tran (M,b1,b1)) |^ m = ZeroMap (V1,V1) ; ::_thesis: verum
end;
supposeA6: len b1 > 0 ; ::_thesis: (Mx2Tran (M,b1,b1)) |^ m = ZeroMap (V1,V1)
A7: dom J = dom (Len J) by MATRIXJ1:def_3;
A8: ( len M = len b1 & len M = Sum (Len J) ) by A1, MATRIX_1:24;
A9: now__::_thesis:_for_x_being_set_st_x_in_dom_b1_holds_
(Z_*_b1)_._x_=_(((Mx2Tran_(M,b1,b1))_|^_m)_*_b1)_._x
let x be set ; ::_thesis: ( x in dom b1 implies (Z * b1) . x = (((Mx2Tran (M,b1,b1)) |^ m) * b1) . x )
assume A10: x in dom b1 ; ::_thesis: (Z * b1) . x = (((Mx2Tran (M,b1,b1)) |^ m) * b1) . x
reconsider n = x as Element of NAT by A10;
set mm = min ((Len J),n);
A11: n in Seg (Sum (Len J)) by A8, A10, FINSEQ_1:def_3;
then A12: min ((Len J),n) in dom (Len J) by MATRIXJ1:def_1;
then A13: (Len J) . (min ((Len J),n)) = len (J . (min ((Len J),n))) by MATRIXJ1:def_3;
A14: (Len J) | (min ((Len J),n)) = Len (J | (min ((Len J),n))) by MATRIXJ1:17;
A15: now__::_thesis:_for_k_being_Nat_st_n_+_k_<=_Sum_(Len_(J_|_(min_((Len_J),n))))_holds_
n_+_k_in_dom_b1
min ((Len J),n) <= len (Len J) by A12, FINSEQ_3:25;
then Sum (Len (J | (min ((Len J),n)))) <= Sum ((Len J) | (len (Len J))) by A14, POLYNOM3:18;
then A16: Sum (Len (J | (min ((Len J),n)))) <= len b1 by A8, FINSEQ_1:58;
let k be Nat; ::_thesis: ( n + k <= Sum (Len (J | (min ((Len J),n)))) implies n + k in dom b1 )
assume n + k <= Sum (Len (J | (min ((Len J),n)))) ; ::_thesis: n + k in dom b1
then A17: n + k <= len b1 by A16, XXREAL_0:2;
( 1 <= n & n <= n + k ) by A11, FINSEQ_1:1, NAT_1:11;
then 1 <= n + k by XXREAL_0:2;
hence n + k in dom b1 by A17, FINSEQ_3:25; ::_thesis: verum
end;
defpred S1[ Nat] means ((Mx2Tran (M,b1,b1)) |^ ($1 + 1)) . (b1 /. n) = 0. V1;
defpred S2[ Nat] means ( n + $1 < Sum (Len (J | (min ((Len J),n)))) implies ((Mx2Tran (M,b1,b1)) |^ ($1 + 1)) . (b1 /. n) = b1 /. ((n + $1) + 1) );
set Sm = Sum ((Len J) | ((min ((Len J),n)) -' 1));
A18: (Len J) . (min ((Len J),n)) = (Len J) /. (min ((Len J),n)) by A12, PARTFUN1:def_6;
(min ((Len J),n)) -' 1 = (min ((Len J),n)) - 1 by A11, MATRIXJ1:7;
then ((min ((Len J),n)) -' 1) + 1 = min ((Len J),n) ;
then (Len J) | (min ((Len J),n)) = ((Len J) | ((min ((Len J),n)) -' 1)) ^ <*((Len J) . (min ((Len J),n)))*> by A12, FINSEQ_5:10;
then A19: Sum (Len (J | (min ((Len J),n)))) = (Sum ((Len J) | ((min ((Len J),n)) -' 1))) + (len (J . (min ((Len J),n)))) by A14, A13, RVSUM_1:74;
A20: Sum ((Len J) | ((min ((Len J),n)) -' 1)) < n by A11, MATRIXJ1:7;
then A21: n -' (Sum ((Len J) | ((min ((Len J),n)) -' 1))) = n - (Sum ((Len J) | ((min ((Len J),n)) -' 1))) by XREAL_1:233;
then A22: n -' (Sum ((Len J) | ((min ((Len J),n)) -' 1))) <> 0 by A11, MATRIXJ1:7;
A23: now__::_thesis:_for_k_being_Nat_st_n_+_k_<=_Sum_(Len_(J_|_(min_((Len_J),n))))_holds_
min_((Len_J),(n_+_k))_=_min_((Len_J),n)
let k be Nat; ::_thesis: ( n + k <= Sum (Len (J | (min ((Len J),n)))) implies min ((Len J),(n + k)) = min ((Len J),n) )
assume n + k <= Sum (Len (J | (min ((Len J),n)))) ; ::_thesis: min ((Len J),(n + k)) = min ((Len J),n)
then A24: (n + k) - (Sum ((Len J) | ((min ((Len J),n)) -' 1))) <= ((Sum ((Len J) | ((min ((Len J),n)) -' 1))) + (len (J . (min ((Len J),n))))) - (Sum ((Len J) | ((min ((Len J),n)) -' 1))) by A19, XREAL_1:9;
1 <= (n -' (Sum ((Len J) | ((min ((Len J),n)) -' 1)))) + k by A22, NAT_1:14;
then (n -' (Sum ((Len J) | ((min ((Len J),n)) -' 1)))) + k in Seg ((Len J) /. (min ((Len J),n))) by A18, A21, A13, A24;
then min ((Len J),(((n -' (Sum ((Len J) | ((min ((Len J),n)) -' 1)))) + k) + (Sum ((Len J) | ((min ((Len J),n)) -' 1))))) = min ((Len J),n) by A12, MATRIXJ1:10;
hence min ((Len J),(n + k)) = min ((Len J),n) by A21; ::_thesis: verum
end;
A25: for k being Nat st S2[k] holds
S2[k + 1]
proof
let k be Nat; ::_thesis: ( S2[k] implies S2[k + 1] )
assume A26: S2[k] ; ::_thesis: S2[k + 1]
set k1 = k + 1;
A27: ( the carrier of V1 = dom ((Mx2Tran (M,b1,b1)) |^ (k + 1)) & n + (k + 1) = (n + k) + 1 ) by FUNCT_2:def_1;
assume A28: n + (k + 1) < Sum (Len (J | (min ((Len J),n)))) ; ::_thesis: ((Mx2Tran (M,b1,b1)) |^ ((k + 1) + 1)) . (b1 /. n) = b1 /. ((n + (k + 1)) + 1)
then ( n + (k + 1) < Sum (Len (J | (min ((Len J),(n + (k + 1)))))) & n + (k + 1) in dom b1 ) by A15, A23;
then A29: (Mx2Tran (M,b1,b1)) . (b1 /. (n + (k + 1))) = ((0. K) * (b1 /. (n + (k + 1)))) + (b1 /. ((n + (k + 1)) + 1)) by A1, Th24
.= (0. V1) + (b1 /. ((n + (k + 1)) + 1)) by VECTSP_1:14
.= b1 /. ((n + (k + 1)) + 1) by RLVECT_1:def_4 ;
thus ((Mx2Tran (M,b1,b1)) |^ ((k + 1) + 1)) . (b1 /. n) = (((Mx2Tran (M,b1,b1)) |^ 1) * ((Mx2Tran (M,b1,b1)) |^ (k + 1))) . (b1 /. n) by VECTSP11:20
.= ((Mx2Tran (M,b1,b1)) |^ 1) . (b1 /. (n + (k + 1))) by A26, A28, A27, FUNCT_1:13, NAT_1:13
.= b1 /. ((n + (k + 1)) + 1) by A29, VECTSP11:19 ; ::_thesis: verum
end;
n <= Sum ((Len J) | (min ((Len J),n))) by A11, MATRIXJ1:def_1;
then A30: (Sum (Len (J | (min ((Len J),n))))) -' n = (Sum (Len (J | (min ((Len J),n))))) - n by A14, XREAL_1:233;
A31: S2[ 0 ]
proof
assume n + 0 < Sum (Len (J | (min ((Len J),n)))) ; ::_thesis: ((Mx2Tran (M,b1,b1)) |^ (0 + 1)) . (b1 /. n) = b1 /. ((n + 0) + 1)
then (Mx2Tran (M,b1,b1)) . (b1 /. n) = ((0. K) * (b1 /. n)) + (b1 /. (n + 1)) by A1, A10, Th24
.= (0. V1) + (b1 /. (n + 1)) by VECTSP_1:14
.= b1 /. (n + 1) by RLVECT_1:def_4 ;
hence ((Mx2Tran (M,b1,b1)) |^ (0 + 1)) . (b1 /. n) = b1 /. ((n + 0) + 1) by VECTSP11:19; ::_thesis: verum
end;
A32: for k being Nat holds S2[k] from NAT_1:sch_2(A31, A25);
A33: S1[(Sum (Len (J | (min ((Len J),n))))) -' n]
proof
percases ( (Sum (Len (J | (min ((Len J),n))))) -' n = 0 or (Sum (Len (J | (min ((Len J),n))))) -' n > 0 ) ;
supposeA34: (Sum (Len (J | (min ((Len J),n))))) -' n = 0 ; ::_thesis: S1[(Sum (Len (J | (min ((Len J),n))))) -' n]
then (Mx2Tran (M,b1,b1)) . (b1 /. n) = (0. K) * (b1 /. n) by A1, A10, A30, Th24
.= 0. V1 by VECTSP_1:14 ;
hence S1[(Sum (Len (J | (min ((Len J),n))))) -' n] by A34, VECTSP11:19; ::_thesis: verum
end;
suppose (Sum (Len (J | (min ((Len J),n))))) -' n > 0 ; ::_thesis: S1[(Sum (Len (J | (min ((Len J),n))))) -' n]
then reconsider S1 = ((Sum (Len (J | (min ((Len J),n))))) -' n) - 1 as Element of NAT by NAT_1:20;
A35: the carrier of V1 = dom ((Mx2Tran (M,b1,b1)) |^ ((Sum (Len (J | (min ((Len J),n))))) -' n)) by FUNCT_2:def_1;
(Sum (Len (J | (min ((Len J),n))))) - 1 < (Sum (Len (J | (min ((Len J),n))))) - 0 by XREAL_1:10;
then A36: ((Mx2Tran (M,b1,b1)) |^ (S1 + 1)) . (b1 /. n) = b1 /. ((n + S1) + 1) by A30, A32
.= b1 /. (Sum (Len (J | (min ((Len J),n))))) by A30 ;
((Sum (Len (J | (min ((Len J),n))))) -' n) + n = Sum (Len (J | (min ((Len J),n)))) by A30;
then ( Sum (Len (J | (min ((Len J),n)))) in dom b1 & min ((Len J),(Sum (Len (J | (min ((Len J),n)))))) = min ((Len J),n) ) by A15, A23;
then A37: (Mx2Tran (M,b1,b1)) . (b1 /. (Sum (Len (J | (min ((Len J),n)))))) = (0. K) * (b1 /. (Sum (Len (J | (min ((Len J),n)))))) by A1, Th24
.= 0. V1 by VECTSP_1:14 ;
thus ((Mx2Tran (M,b1,b1)) |^ (((Sum (Len (J | (min ((Len J),n))))) -' n) + 1)) . (b1 /. n) = (((Mx2Tran (M,b1,b1)) |^ 1) * ((Mx2Tran (M,b1,b1)) |^ ((Sum (Len (J | (min ((Len J),n))))) -' n))) . (b1 /. n) by VECTSP11:20
.= ((Mx2Tran (M,b1,b1)) |^ 1) . (((Mx2Tran (M,b1,b1)) |^ ((Sum (Len (J | (min ((Len J),n))))) -' n)) . (b1 /. n)) by A35, FUNCT_1:13
.= 0. V1 by A36, A37, VECTSP11:19 ; ::_thesis: verum
end;
end;
end;
(Sum ((Len J) | ((min ((Len J),n)) -' 1))) - n < n - n by A20, XREAL_1:9;
then A38: (len (J . (min ((Len J),n)))) + ((Sum ((Len J) | ((min ((Len J),n)) -' 1))) - n) < (len (J . (min ((Len J),n)))) + 0 by XREAL_1:6;
then 0 < m by A2, A7, A12, A30, A19;
then reconsider m1 = m - 1 as Element of NAT by NAT_1:20;
len (J . (min ((Len J),n))) <= m by A2, A7, A12;
then (Sum (Len (J | (min ((Len J),n))))) -' n < m1 + 1 by A30, A19, A38, XXREAL_0:2;
then A39: (Sum (Len (J | (min ((Len J),n))))) -' n <= m1 by NAT_1:13;
A40: for k being Nat st (Sum (Len (J | (min ((Len J),n))))) -' n <= k & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( (Sum (Len (J | (min ((Len J),n))))) -' n <= k & S1[k] implies S1[k + 1] )
assume (Sum (Len (J | (min ((Len J),n))))) -' n <= k ; ::_thesis: ( not S1[k] or S1[k + 1] )
set k1 = k + 1;
assume A41: S1[k] ; ::_thesis: S1[k + 1]
A42: dom ((Mx2Tran (M,b1,b1)) |^ (k + 1)) = the carrier of V1 by FUNCT_2:def_1;
thus ((Mx2Tran (M,b1,b1)) |^ ((k + 1) + 1)) . (b1 /. n) = (((Mx2Tran (M,b1,b1)) |^ 1) * ((Mx2Tran (M,b1,b1)) |^ (k + 1))) . (b1 /. n) by VECTSP11:20
.= ((Mx2Tran (M,b1,b1)) |^ 1) . (((Mx2Tran (M,b1,b1)) |^ (k + 1)) . (b1 /. n)) by A42, FUNCT_1:13
.= (Mx2Tran (M,b1,b1)) . (0. V1) by A41, VECTSP11:19
.= (Mx2Tran (M,b1,b1)) . ((0. K) * (0. V1)) by VECTSP_1:14
.= (0. K) * ((Mx2Tran (M,b1,b1)) . (0. V1)) by MOD_2:def_2
.= 0. V1 by VECTSP_1:14 ; ::_thesis: verum
end;
for k being Nat st (Sum (Len (J | (min ((Len J),n))))) -' n <= k holds
S1[k] from NAT_1:sch_8(A33, A40);
then A43: ((Mx2Tran (M,b1,b1)) |^ (m1 + 1)) . (b1 /. n) = 0. V1 by A39;
thus (Z * b1) . x = Z . (b1 . x) by A10, FUNCT_1:13
.= Z . (b1 /. x) by A10, PARTFUN1:def_6
.= ( the carrier of V1 --> (0. V1)) . (b1 /. x) by GRCAT_1:def_7
.= 0. V1 by FUNCOP_1:7
.= ((Mx2Tran (M,b1,b1)) |^ m) . (b1 . n) by A10, A43, PARTFUN1:def_6
.= (((Mx2Tran (M,b1,b1)) |^ m) * b1) . x by A10, FUNCT_1:13 ; ::_thesis: verum
end;
( dom (Z * b1) = dom b1 & dom (((Mx2Tran (M,b1,b1)) |^ m) * b1) = dom b1 ) by A4, A3, RELAT_1:27;
hence (Mx2Tran (M,b1,b1)) |^ m = ZeroMap (V1,V1) by A6, A9, FUNCT_1:2, MATRLIN:22; ::_thesis: verum
end;
end;
end;
Lm3: for n being Nat
for K being Field
for L being Element of K
for V1 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) & len b1 <> 0 holds
( ((Mx2Tran (M,b1,b1)) |^ n) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = ((power K) . (L,n)) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) & Sum (Len (J | (min ((Len J),(len b1))))) in dom b1 )
proof
let n be Nat; ::_thesis: for K being Field
for L being Element of K
for V1 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) & len b1 <> 0 holds
( ((Mx2Tran (M,b1,b1)) |^ n) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = ((power K) . (L,n)) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) & Sum (Len (J | (min ((Len J),(len b1))))) in dom b1 )
let K be Field; ::_thesis: for L being Element of K
for V1 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) & len b1 <> 0 holds
( ((Mx2Tran (M,b1,b1)) |^ n) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = ((power K) . (L,n)) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) & Sum (Len (J | (min ((Len J),(len b1))))) in dom b1 )
let L be Element of K; ::_thesis: for V1 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) & len b1 <> 0 holds
( ((Mx2Tran (M,b1,b1)) |^ n) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = ((power K) . (L,n)) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) & Sum (Len (J | (min ((Len J),(len b1))))) in dom b1 )
let V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) & len b1 <> 0 holds
( ((Mx2Tran (M,b1,b1)) |^ n) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = ((power K) . (L,n)) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) & Sum (Len (J | (min ((Len J),(len b1))))) in dom b1 )
let b1 be OrdBasis of V1; ::_thesis: for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) & len b1 <> 0 holds
( ((Mx2Tran (M,b1,b1)) |^ n) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = ((power K) . (L,n)) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) & Sum (Len (J | (min ((Len J),(len b1))))) in dom b1 )
let J be FinSequence_of_Jordan_block of L,K; ::_thesis: for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) & len b1 <> 0 holds
( ((Mx2Tran (M,b1,b1)) |^ n) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = ((power K) . (L,n)) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) & Sum (Len (J | (min ((Len J),(len b1))))) in dom b1 )
set B = len b1;
set m = min ((Len J),(len b1));
set S = Sum (Len (J | (min ((Len J),(len b1)))));
A1: Seg (len b1) = dom b1 by FINSEQ_1:def_3;
let M be Matrix of len b1, len b1,K; ::_thesis: ( M = block_diagonal (J,(0. K)) & len b1 <> 0 implies ( ((Mx2Tran (M,b1,b1)) |^ n) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = ((power K) . (L,n)) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) & Sum (Len (J | (min ((Len J),(len b1))))) in dom b1 ) )
assume A2: M = block_diagonal (J,(0. K)) ; ::_thesis: ( not len b1 <> 0 or ( ((Mx2Tran (M,b1,b1)) |^ n) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = ((power K) . (L,n)) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) & Sum (Len (J | (min ((Len J),(len b1))))) in dom b1 ) )
A3: ( len b1 = len M & len M = Sum (Len J) ) by A2, MATRIX_1:24;
set MT = Mx2Tran (M,b1,b1);
defpred S1[ Nat] means ((Mx2Tran (M,b1,b1)) |^ $1) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = ((power K) . (L,$1)) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1)))))));
((Mx2Tran (M,b1,b1)) |^ 0) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = (id V1) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) by VECTSP11:18
.= b1 /. (Sum (Len (J | (min ((Len J),(len b1)))))) by FUNCT_1:18
.= (1_ K) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) by VECTSP_1:def_17
.= ((power K) . (L,0)) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) by GROUP_1:def_7 ;
then A4: S1[ 0 ] ;
A5: ( Sum ((Len J) | (min ((Len J),(len b1)))) = Sum (Len (J | (min ((Len J),(len b1))))) & (Len J) | (len (Len J)) = Len J ) by FINSEQ_1:58, MATRIXJ1:17;
assume len b1 <> 0 ; ::_thesis: ( ((Mx2Tran (M,b1,b1)) |^ n) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = ((power K) . (L,n)) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) & Sum (Len (J | (min ((Len J),(len b1))))) in dom b1 )
then A6: len b1 in Seg (len b1) by FINSEQ_1:3;
then min ((Len J),(len b1)) in dom (Len J) by A3, MATRIXJ1:def_1;
then min ((Len J),(len b1)) <= len (Len J) by FINSEQ_3:25;
then A7: Sum ((Len J) | (min ((Len J),(len b1)))) <= Sum ((Len J) | (len (Len J))) by POLYNOM3:18;
A8: len b1 <= Sum ((Len J) | (min ((Len J),(len b1)))) by A3, A6, MATRIXJ1:def_1;
then len b1 = Sum (Len (J | (min ((Len J),(len b1))))) by A3, A7, A5, XXREAL_0:1;
then A9: (Mx2Tran (M,b1,b1)) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = L * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) by A2, A6, A1, Th24;
A10: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A11: S1[n] ; ::_thesis: S1[n + 1]
A12: dom ((Mx2Tran (M,b1,b1)) |^ n) = the carrier of V1 by FUNCT_2:def_1;
A13: n in NAT by ORDINAL1:def_12;
thus ((Mx2Tran (M,b1,b1)) |^ (n + 1)) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = (((Mx2Tran (M,b1,b1)) |^ 1) * ((Mx2Tran (M,b1,b1)) |^ n)) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) by VECTSP11:20
.= ((Mx2Tran (M,b1,b1)) * ((Mx2Tran (M,b1,b1)) |^ n)) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) by VECTSP11:19
.= (Mx2Tran (M,b1,b1)) . (((power K) . (L,n)) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1)))))))) by A11, A12, FUNCT_1:13
.= ((power K) . (L,n)) * (L * (b1 /. (Sum (Len (J | (min ((Len J),(len b1)))))))) by A9, MOD_2:def_2
.= (((power K) . (L,n)) * L) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) by VECTSP_1:def_16
.= ((power K) . (L,(n + 1))) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) by A13, GROUP_1:def_7 ; ::_thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch_2(A4, A10);
hence ( ((Mx2Tran (M,b1,b1)) |^ n) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = ((power K) . (L,n)) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) & Sum (Len (J | (min ((Len J),(len b1))))) in dom b1 ) by A3, A6, A1, A8, A7, A5, XXREAL_0:1; ::_thesis: verum
end;
theorem :: MATRIXJ2:26
for K being Field
for L being Element of K
for V1 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) holds
( Mx2Tran (M,b1,b1) is nilpotent iff ( len b1 = 0 or L = 0. K ) )
proof
let K be Field; ::_thesis: for L being Element of K
for V1 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) holds
( Mx2Tran (M,b1,b1) is nilpotent iff ( len b1 = 0 or L = 0. K ) )
let L be Element of K; ::_thesis: for V1 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) holds
( Mx2Tran (M,b1,b1) is nilpotent iff ( len b1 = 0 or L = 0. K ) )
let V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) holds
( Mx2Tran (M,b1,b1) is nilpotent iff ( len b1 = 0 or L = 0. K ) )
let b1 be OrdBasis of V1; ::_thesis: for J being FinSequence_of_Jordan_block of L,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) holds
( Mx2Tran (M,b1,b1) is nilpotent iff ( len b1 = 0 or L = 0. K ) )
let J be FinSequence_of_Jordan_block of L,K; ::_thesis: for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) holds
( Mx2Tran (M,b1,b1) is nilpotent iff ( len b1 = 0 or L = 0. K ) )
let M be Matrix of len b1, len b1,K; ::_thesis: ( M = block_diagonal (J,(0. K)) implies ( Mx2Tran (M,b1,b1) is nilpotent iff ( len b1 = 0 or L = 0. K ) ) )
assume A1: M = block_diagonal (J,(0. K)) ; ::_thesis: ( Mx2Tran (M,b1,b1) is nilpotent iff ( len b1 = 0 or L = 0. K ) )
set MT = Mx2Tran (M,b1,b1);
thus ( not Mx2Tran (M,b1,b1) is nilpotent or len b1 = 0 or L = 0. K ) ::_thesis: ( ( len b1 = 0 or L = 0. K ) implies Mx2Tran (M,b1,b1) is nilpotent )
proof
set B = len b1;
set m = min ((Len J),(len b1));
set S = Sum (Len (J | (min ((Len J),(len b1)))));
assume Mx2Tran (M,b1,b1) is nilpotent ; ::_thesis: ( len b1 = 0 or L = 0. K )
then reconsider MT = Mx2Tran (M,b1,b1) as nilpotent linear-transformation of V1,V1 ;
rng b1 is Basis of V1 by MATRLIN:def_2;
then A2: rng b1 is linearly-independent Subset of V1 by VECTSP_7:def_3;
assume A3: len b1 <> 0 ; ::_thesis: L = 0. K
then Sum (Len (J | (min ((Len J),(len b1))))) in dom b1 by A1, Lm3;
then ( b1 . (Sum (Len (J | (min ((Len J),(len b1)))))) in rng b1 & b1 /. (Sum (Len (J | (min ((Len J),(len b1)))))) = b1 . (Sum (Len (J | (min ((Len J),(len b1)))))) ) by FUNCT_1:def_3, PARTFUN1:def_6;
then A4: b1 /. (Sum (Len (J | (min ((Len J),(len b1)))))) <> 0. V1 by A2, VECTSP_7:2;
((power K) . (L,(deg MT))) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) = (MT |^ (deg MT)) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) by A1, A3, Lm3
.= (ZeroMap (V1,V1)) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) by Def5
.= ( the carrier of V1 --> (0. V1)) . (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) by GRCAT_1:def_7
.= 0. V1 by FUNCOP_1:7
.= (0. K) * (b1 /. (Sum (Len (J | (min ((Len J),(len b1))))))) by VECTSP_1:14 ;
then 0. K = (power K) . (L,(deg MT)) by A4, VECTSP10:4
.= Product ((deg MT) |-> L) by MATRIXJ1:5 ;
then A5: ex k being Element of NAT st
( k in dom ((deg MT) |-> L) & ((deg MT) |-> L) . k = 0. K ) by FVSUM_1:82;
dom ((deg MT) |-> L) = Seg (deg MT) by FINSEQ_2:124;
hence L = 0. K by A5, FINSEQ_2:57; ::_thesis: verum
end;
assume A6: ( len b1 = 0 or L = 0. K ) ; ::_thesis: Mx2Tran (M,b1,b1) is nilpotent
percases ( len b1 = 0 or L = 0. K ) by A6;
suppose len b1 = 0 ; ::_thesis: Mx2Tran (M,b1,b1) is nilpotent
then dim V1 = 0 by MATRLIN2:21;
then (Omega). V1 = (0). V1 by VECTSP_9:29;
then A7: the carrier of V1 = {(0. V1)} by VECTSP_4:def_3;
rng ((Mx2Tran (M,b1,b1)) |^ 1) c= the carrier of V1 by RELAT_1:def_19;
then rng ((Mx2Tran (M,b1,b1)) |^ 1) = {(0. V1)} by A7, ZFMISC_1:33;
then (Mx2Tran (M,b1,b1)) |^ 1 = (dom ((Mx2Tran (M,b1,b1)) |^ 1)) --> (0. V1) by FUNCOP_1:9
.= the carrier of V1 --> (0. V1) by FUNCT_2:def_1
.= ZeroMap (V1,V1) by GRCAT_1:def_7 ;
hence Mx2Tran (M,b1,b1) is nilpotent by Th13; ::_thesis: verum
end;
supposeA8: L = 0. K ; ::_thesis: Mx2Tran (M,b1,b1) is nilpotent
now__::_thesis:_for_i_being_Nat_st_i_in_dom_J_holds_
len_(J_._i)_<=_Sum_(Len_J)
A9: dom J = dom (Len J) by MATRIXJ1:def_3;
let i be Nat; ::_thesis: ( i in dom J implies len (J . i) <= Sum (Len J) )
assume A10: i in dom J ; ::_thesis: len (J . i) <= Sum (Len J)
len (J . i) = (Len J) . i by A10, A9, MATRIXJ1:def_3;
hence len (J . i) <= Sum (Len J) by A10, A9, POLYNOM3:4; ::_thesis: verum
end;
then (Mx2Tran (M,b1,b1)) |^ (Sum (Len J)) = ZeroMap (V1,V1) by A1, A8, Th25;
hence Mx2Tran (M,b1,b1) is nilpotent by Th13; ::_thesis: verum
end;
end;
end;
theorem :: MATRIXJ2:27
for K being Field
for V1 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of 0. K,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) & len b1 > 0 holds
for F being nilpotent Function of V1,V1 st F = Mx2Tran (M,b1,b1) holds
( ex i being Nat st
( i in dom J & len (J . i) = deg F ) & ( for i being Nat st i in dom J holds
len (J . i) <= deg F ) )
proof
let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of 0. K,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) & len b1 > 0 holds
for F being nilpotent Function of V1,V1 st F = Mx2Tran (M,b1,b1) holds
( ex i being Nat st
( i in dom J & len (J . i) = deg F ) & ( for i being Nat st i in dom J holds
len (J . i) <= deg F ) )
let V1 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1
for J being FinSequence_of_Jordan_block of 0. K,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) & len b1 > 0 holds
for F being nilpotent Function of V1,V1 st F = Mx2Tran (M,b1,b1) holds
( ex i being Nat st
( i in dom J & len (J . i) = deg F ) & ( for i being Nat st i in dom J holds
len (J . i) <= deg F ) )
let b1 be OrdBasis of V1; ::_thesis: for J being FinSequence_of_Jordan_block of 0. K,K
for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) & len b1 > 0 holds
for F being nilpotent Function of V1,V1 st F = Mx2Tran (M,b1,b1) holds
( ex i being Nat st
( i in dom J & len (J . i) = deg F ) & ( for i being Nat st i in dom J holds
len (J . i) <= deg F ) )
let J be FinSequence_of_Jordan_block of 0. K,K; ::_thesis: for M being Matrix of len b1, len b1,K st M = block_diagonal (J,(0. K)) & len b1 > 0 holds
for F being nilpotent Function of V1,V1 st F = Mx2Tran (M,b1,b1) holds
( ex i being Nat st
( i in dom J & len (J . i) = deg F ) & ( for i being Nat st i in dom J holds
len (J . i) <= deg F ) )
let M be Matrix of len b1, len b1,K; ::_thesis: ( M = block_diagonal (J,(0. K)) & len b1 > 0 implies for F being nilpotent Function of V1,V1 st F = Mx2Tran (M,b1,b1) holds
( ex i being Nat st
( i in dom J & len (J . i) = deg F ) & ( for i being Nat st i in dom J holds
len (J . i) <= deg F ) ) )
assume that
A1: M = block_diagonal (J,(0. K)) and
A2: len b1 > 0 ; ::_thesis: for F being nilpotent Function of V1,V1 st F = Mx2Tran (M,b1,b1) holds
( ex i being Nat st
( i in dom J & len (J . i) = deg F ) & ( for i being Nat st i in dom J holds
len (J . i) <= deg F ) )
A3: ( len M = len b1 & len M = Sum (Len J) ) by A1, MATRIX_1:def_2;
defpred S1[ Nat] means for i being Nat st i in dom J holds
len (J . i) <= $1;
set mm = min ((Len J),(len b1));
A4: dom J = dom (Len J) by MATRIXJ1:def_3;
now__::_thesis:_for_i_being_Nat_st_i_in_dom_J_holds_
len_(J_._i)_<=_Sum_(Len_J)
let i be Nat; ::_thesis: ( i in dom J implies len (J . i) <= Sum (Len J) )
assume A5: i in dom J ; ::_thesis: len (J . i) <= Sum (Len J)
len (J . i) = (Len J) . i by A4, A5, MATRIXJ1:def_3;
hence len (J . i) <= Sum (Len J) by A4, A5, POLYNOM3:4; ::_thesis: verum
end;
then A6: ex k being Nat st S1[k] ;
consider MIN being Nat such that
A7: S1[MIN] and
A8: for m being Nat st S1[m] holds
MIN <= m from NAT_1:sch_5(A6);
len b1 in Seg (len b1) by A2, FINSEQ_1:3;
then A9: min ((Len J),(len b1)) in dom (Len J) by A3, MATRIXJ1:def_1;
A10: ex i being Nat st
( i in dom J & len (J . i) = MIN )
proof
assume A11: for i being Nat st i in dom J holds
len (J . i) <> MIN ; ::_thesis: contradiction
len (J . (min ((Len J),(len b1)))) <= MIN by A9, A4, A7;
then len (J . (min ((Len J),(len b1)))) < MIN by A9, A4, A11, XXREAL_0:1;
then reconsider M1 = MIN - 1 as Element of NAT by NAT_1:20;
now__::_thesis:_for_i_being_Nat_st_i_in_dom_J_holds_
len_(J_._i)_<=_M1
let i be Nat; ::_thesis: ( i in dom J implies len (J . i) <= M1 )
assume A12: i in dom J ; ::_thesis: len (J . i) <= M1
len (J . i) <= MIN by A7, A12;
then len (J . i) < M1 + 1 by A11, A12, XXREAL_0:1;
hence len (J . i) <= M1 by NAT_1:13; ::_thesis: verum
end;
then M1 + 1 <= M1 by A8;
hence contradiction by NAT_1:13; ::_thesis: verum
end;
A13: (Len J) | (len (Len J)) = Len J by FINSEQ_1:58;
let F be nilpotent Function of V1,V1; ::_thesis: ( F = Mx2Tran (M,b1,b1) implies ( ex i being Nat st
( i in dom J & len (J . i) = deg F ) & ( for i being Nat st i in dom J holds
len (J . i) <= deg F ) ) )
assume A14: F = Mx2Tran (M,b1,b1) ; ::_thesis: ( ex i being Nat st
( i in dom J & len (J . i) = deg F ) & ( for i being Nat st i in dom J holds
len (J . i) <= deg F ) )
consider i being Nat such that
A15: i in dom J and
A16: len (J . i) = MIN by A10;
A17: (Len J) . i = (Len J) /. i by A4, A15, PARTFUN1:def_6;
set S = Sum ((Len J) | (i -' 1));
defpred S2[ Nat] means ( $1 in Seg MIN & $1 <> MIN implies (F |^ $1) . (b1 /. ((Sum ((Len J) | (i -' 1))) + 1)) = b1 /. (((Sum ((Len J) | (i -' 1))) + $1) + 1) );
A18: len (J . i) = (Len J) . i by A4, A15, MATRIXJ1:def_3;
i <= len (Len J) by A4, A15, FINSEQ_3:25;
then Sum ((Len J) | i) <= Sum ((Len J) | (len (Len J))) by A15, POLYNOM3:18;
then A19: ( dom b1 = Seg (len b1) & Seg (Sum ((Len J) | i)) c= Seg (Sum (Len J)) ) by A13, FINSEQ_1:5, FINSEQ_1:def_3;
1 <= i by A15, FINSEQ_3:25;
then i -' 1 = i - 1 by XREAL_1:233;
then A20: i = (i -' 1) + 1 ;
A21: for n being Nat st S2[n] holds
S2[n + 1]
proof
(Len J) | i = ((Len J) | (i -' 1)) ^ <*MIN*> by A4, A15, A16, A18, A20, FINSEQ_5:10;
then A22: Sum ((Len J) | i) = (Sum ((Len J) | (i -' 1))) + MIN by RVSUM_1:74;
let n be Nat; ::_thesis: ( S2[n] implies S2[n + 1] )
assume A23: S2[n] ; ::_thesis: S2[n + 1]
A24: (Len J) | i = Len (J | i) by MATRIXJ1:17;
set n1 = n + 1;
assume that
A25: n + 1 in Seg MIN and
A26: n + 1 <> MIN ; ::_thesis: (F |^ (n + 1)) . (b1 /. ((Sum ((Len J) | (i -' 1))) + 1)) = b1 /. (((Sum ((Len J) | (i -' 1))) + (n + 1)) + 1)
A27: n + 1 <= MIN by A25, FINSEQ_1:1;
then n + 1 < MIN by A26, XXREAL_0:1;
then A28: (Sum ((Len J) | (i -' 1))) + (n + 1) < Sum ((Len J) | i) by A22, XREAL_1:6;
( (Sum ((Len J) | (i -' 1))) + (n + 1) in Seg (Sum ((Len J) | i)) & min ((Len J),((Sum ((Len J) | (i -' 1))) + (n + 1))) = i ) by A4, A15, A16, A18, A17, A25, MATRIXJ1:10;
then A29: F . (b1 /. ((Sum ((Len J) | (i -' 1))) + (n + 1))) = ((0. K) * (b1 /. ((Sum ((Len J) | (i -' 1))) + (n + 1)))) + (b1 /. (((Sum ((Len J) | (i -' 1))) + (n + 1)) + 1)) by A1, A3, A14, A19, A28, A24, Th24
.= (0. V1) + (b1 /. (((Sum ((Len J) | (i -' 1))) + (n + 1)) + 1)) by VECTSP_1:14
.= b1 /. (((Sum ((Len J) | (i -' 1))) + (n + 1)) + 1) by RLVECT_1:def_4 ;
A30: n < MIN by A27, NAT_1:13;
now__::_thesis:_(F_|^_(n_+_1))_._(b1_/._((Sum_((Len_J)_|_(i_-'_1)))_+_1))_=_b1_/._(((Sum_((Len_J)_|_(i_-'_1)))_+_(n_+_1))_+_1)
percases ( n = 0 or n >= 1 ) by NAT_1:14;
suppose n = 0 ; ::_thesis: (F |^ (n + 1)) . (b1 /. ((Sum ((Len J) | (i -' 1))) + 1)) = b1 /. (((Sum ((Len J) | (i -' 1))) + (n + 1)) + 1)
hence (F |^ (n + 1)) . (b1 /. ((Sum ((Len J) | (i -' 1))) + 1)) = b1 /. (((Sum ((Len J) | (i -' 1))) + (n + 1)) + 1) by A29, VECTSP11:19; ::_thesis: verum
end;
supposeA31: n >= 1 ; ::_thesis: (F |^ (n + 1)) . (b1 /. ((Sum ((Len J) | (i -' 1))) + 1)) = b1 /. (((Sum ((Len J) | (i -' 1))) + (n + 1)) + 1)
A32: dom (F |^ n) = the carrier of V1 by FUNCT_2:def_1;
thus (F |^ (n + 1)) . (b1 /. ((Sum ((Len J) | (i -' 1))) + 1)) = ((F |^ 1) * (F |^ n)) . (b1 /. ((Sum ((Len J) | (i -' 1))) + 1)) by VECTSP11:20
.= (F |^ 1) . ((F |^ n) . (b1 /. ((Sum ((Len J) | (i -' 1))) + 1))) by A32, FUNCT_1:13
.= b1 /. (((Sum ((Len J) | (i -' 1))) + (n + 1)) + 1) by A23, A30, A29, A31, FINSEQ_1:1, VECTSP11:19 ; ::_thesis: verum
end;
end;
end;
hence (F |^ (n + 1)) . (b1 /. ((Sum ((Len J) | (i -' 1))) + 1)) = b1 /. (((Sum ((Len J) | (i -' 1))) + (n + 1)) + 1) ; ::_thesis: verum
end;
A33: S2[ 0 ] by FINSEQ_1:1;
A34: for n being Nat holds S2[n] from NAT_1:sch_2(A33, A21);
A35: deg F >= MIN
proof
set D = deg F;
rng b1 is Basis of V1 by MATRLIN:def_2;
then A36: rng b1 is linearly-independent Subset of V1 by VECTSP_7:def_3;
assume A37: deg F < MIN ; ::_thesis: contradiction
then ( 1 <= 1 + (deg F) & (deg F) + 1 <= MIN ) by NAT_1:11, NAT_1:13;
then (deg F) + 1 in Seg MIN ;
then (Sum ((Len J) | (i -' 1))) + ((deg F) + 1) in Seg (Sum ((Len J) | i)) by A4, A15, A16, A18, A17, MATRIXJ1:10;
then A38: ( b1 /. (((Sum ((Len J) | (i -' 1))) + (deg F)) + 1) = b1 . (((Sum ((Len J) | (i -' 1))) + (deg F)) + 1) & b1 . (((Sum ((Len J) | (i -' 1))) + (deg F)) + 1) in rng b1 ) by A3, A19, FUNCT_1:def_3, PARTFUN1:def_6;
deg F <> 0
proof
assume deg F = 0 ; ::_thesis: contradiction
then [#] V1 = {(0. V1)} by Th15;
then (Omega). V1 = (0). V1 by VECTSP_4:def_3;
then dim V1 = 0 by VECTSP_9:29;
hence contradiction by A2, MATRLIN2:21; ::_thesis: verum
end;
then deg F >= 1 by NAT_1:14;
then deg F in Seg MIN by A37, FINSEQ_1:1;
then b1 /. (((Sum ((Len J) | (i -' 1))) + (deg F)) + 1) = (F |^ (deg F)) . (b1 /. ((Sum ((Len J) | (i -' 1))) + 1)) by A34, A37
.= (ZeroMap (V1,V1)) . (b1 /. ((Sum ((Len J) | (i -' 1))) + 1)) by Def5
.= ( the carrier of V1 --> (0. V1)) . (b1 /. ((Sum ((Len J) | (i -' 1))) + 1)) by GRCAT_1:def_7
.= 0. V1 by FUNCOP_1:7 ;
hence contradiction by A38, A36, VECTSP_7:2; ::_thesis: verum
end;
F |^ MIN = ZeroMap (V1,V1) by A1, A14, A7, Th25;
then deg F <= MIN by Def5;
then deg F = MIN by A35, XXREAL_0:1;
hence ( ex i being Nat st
( i in dom J & len (J . i) = deg F ) & ( for i being Nat st i in dom J holds
len (J . i) <= deg F ) ) by A7, A10; ::_thesis: verum
end;
Lm4: for K being Field
for V1, V2 being VectSp of K
for f being linear-transformation of V1,V2
for W1 being Subspace of V1
for W2 being Subspace of V2
for F being Function of W1,W2 st F = f | W1 holds
F is linear-transformation of W1,W2
proof
let K be Field; ::_thesis: for V1, V2 being VectSp of K
for f being linear-transformation of V1,V2
for W1 being Subspace of V1
for W2 being Subspace of V2
for F being Function of W1,W2 st F = f | W1 holds
F is linear-transformation of W1,W2
let V1, V2 be VectSp of K; ::_thesis: for f being linear-transformation of V1,V2
for W1 being Subspace of V1
for W2 being Subspace of V2
for F being Function of W1,W2 st F = f | W1 holds
F is linear-transformation of W1,W2
let f be linear-transformation of V1,V2; ::_thesis: for W1 being Subspace of V1
for W2 being Subspace of V2
for F being Function of W1,W2 st F = f | W1 holds
F is linear-transformation of W1,W2
let W1 be Subspace of V1; ::_thesis: for W2 being Subspace of V2
for F being Function of W1,W2 st F = f | W1 holds
F is linear-transformation of W1,W2
let W2 be Subspace of V2; ::_thesis: for F being Function of W1,W2 st F = f | W1 holds
F is linear-transformation of W1,W2
let F be Function of W1,W2; ::_thesis: ( F = f | W1 implies F is linear-transformation of W1,W2 )
assume A1: F = f | W1 ; ::_thesis: F is linear-transformation of W1,W2
A2: now__::_thesis:_for_a_being_Scalar_of_K
for_w_being_Vector_of_W1_holds_F_._(a_*_w)_=_a_*_(F_._w)
let a be Scalar of K; ::_thesis: for w being Vector of W1 holds F . (a * w) = a * (F . w)
let w be Vector of W1; ::_thesis: F . (a * w) = a * (F . w)
thus F . (a * w) = a * ((f | W1) . w) by A1, MOD_2:def_2
.= a * (F . w) by A1, VECTSP_4:14 ; ::_thesis: verum
end;
now__::_thesis:_for_w1,_w2_being_Vector_of_W1_holds_F_._(w1_+_w2)_=_(F_._w1)_+_(F_._w2)
let w1, w2 be Vector of W1; ::_thesis: F . (w1 + w2) = (F . w1) + (F . w2)
thus F . (w1 + w2) = ((f | W1) . w1) + ((f | W1) . w2) by A1, VECTSP_1:def_20
.= (F . w1) + (F . w2) by A1, VECTSP_4:13 ; ::_thesis: verum
end;
then ( F is additive & F is homogeneous ) by A2, VECTSP_1:def_20, MOD_2:def_2;
hence F is linear-transformation of W1,W2 ; ::_thesis: verum
end;
theorem Th28: :: MATRIXJ2:28
for K being Field
for V1, V2 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for b2 being OrdBasis of V2
for L being Element of K st len b1 = len b2 holds
for F being linear-transformation of V1,V2 st ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
proof
let K be Field; ::_thesis: for V1, V2 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for b2 being OrdBasis of V2
for L being Element of K st len b1 = len b2 holds
for F being linear-transformation of V1,V2 st ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
defpred S1[ Nat] means for V1, V2 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for b2 being OrdBasis of V2
for L being Element of K st len b1 = len b2 holds
for F being linear-transformation of V1,V2 st $1 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } & ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K));
A1: S1[ 0 ]
proof
let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1
for b2 being OrdBasis of V2
for L being Element of K st len b1 = len b2 holds
for F being linear-transformation of V1,V2 st 0 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } & ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2
for L being Element of K st len b1 = len b2 holds
for F being linear-transformation of V1,V2 st 0 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } & ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
let b2 be OrdBasis of V2; ::_thesis: for L being Element of K st len b1 = len b2 holds
for F being linear-transformation of V1,V2 st 0 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } & ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
let L be Element of K; ::_thesis: ( len b1 = len b2 implies for F being linear-transformation of V1,V2 st 0 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } & ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K)) )
assume len b1 = len b2 ; ::_thesis: for F being linear-transformation of V1,V2 st 0 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } & ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
set Lb = len b1;
reconsider J = {} as FinSequence_of_Jordan_block of L,K by Th10;
let F be linear-transformation of V1,V2; ::_thesis: ( 0 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } & ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) implies ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K)) )
assume that
A2: 0 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } and
A3: for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ; ::_thesis: ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
reconsider J = J as non-empty FinSequence_of_Jordan_block of L,K ;
take J ; ::_thesis: AutMt (F,b1,b2) = block_diagonal (J,(0. K))
len b1 = 0
proof
assume len b1 <> 0 ; ::_thesis: contradiction
then A4: len b1 in Seg (len b1) by FINSEQ_1:3;
A5: dom b1 = Seg (len b1) by FINSEQ_1:def_3;
percases ( F . (b1 /. (len b1)) = L * (b2 /. (len b1)) or ( (len b1) + 1 in dom b1 & F . (b1 /. (len b1)) = (L * (b2 /. (len b1))) + (b2 /. ((len b1) + 1)) ) ) by A3, A4, A5;
suppose F . (b1 /. (len b1)) = L * (b2 /. (len b1)) ; ::_thesis: contradiction
then len b1 in { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } by A4, A5;
hence contradiction by A2; ::_thesis: verum
end;
suppose ( (len b1) + 1 in dom b1 & F . (b1 /. (len b1)) = (L * (b2 /. (len b1))) + (b2 /. ((len b1) + 1)) ) ; ::_thesis: contradiction
then (len b1) + 1 <= len b1 by FINSEQ_3:25;
hence contradiction by NAT_1:13; ::_thesis: verum
end;
end;
end;
then len (AutMt (F,b1,b2)) = 0 by MATRIX_1:def_2;
then AutMt (F,b1,b2) = {} ;
hence AutMt (F,b1,b2) = block_diagonal (J,(0. K)) by MATRIXJ1:22; ::_thesis: verum
end;
A6: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A7: S1[n] ; ::_thesis: S1[n + 1]
set n1 = n + 1;
let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1
for b2 being OrdBasis of V2
for L being Element of K st len b1 = len b2 holds
for F being linear-transformation of V1,V2 st n + 1 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } & ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2
for L being Element of K st len b1 = len b2 holds
for F being linear-transformation of V1,V2 st n + 1 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } & ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
let b2 be OrdBasis of V2; ::_thesis: for L being Element of K st len b1 = len b2 holds
for F being linear-transformation of V1,V2 st n + 1 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } & ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
let L be Element of K; ::_thesis: ( len b1 = len b2 implies for F being linear-transformation of V1,V2 st n + 1 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } & ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K)) )
assume A8: len b1 = len b2 ; ::_thesis: for F being linear-transformation of V1,V2 st n + 1 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } & ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
A9: dom b1 = dom b2 by A8, FINSEQ_3:29;
set Lb = len b1;
let F be linear-transformation of V1,V2; ::_thesis: ( n + 1 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } & ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) implies ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K)) )
assume that
A10: n + 1 = card { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } and
A11: for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ; ::_thesis: ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
set FF = { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } ;
reconsider FF = { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } as finite set by A10;
consider x being set such that
A12: x in FF by A10, CARD_1:27, XBOOLE_0:def_1;
ex ii being Element of NAT st
( ii = x & ii in dom b1 & F . (b1 /. ii) = L * (b2 /. ii) ) by A12;
then dom b1 <> {} ;
then Seg (len b1) <> {} by FINSEQ_1:def_3;
then A13: len b1 <> 0 ;
A14: dom b1 = Seg (len b1) by FINSEQ_1:def_3;
then A15: not (len b1) + 1 in dom b1 by FINSEQ_3:8;
A16: len b1 in dom b1 by A14, A13, FINSEQ_1:3;
then F . (b1 /. (len b1)) = L * (b2 /. (len b1)) by A11, A15;
then A17: len b1 in FF by A16;
percases ( n = 0 or n <> 0 ) ;
supposeA18: n = 0 ; ::_thesis: ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
set J = Jordan_block (L,(len b1));
reconsider JJ = <*(Jordan_block (L,(len b1)))*> as FinSequence_of_Jordan_block of L,K by Th11;
now__::_thesis:_for_x_being_set_st_x_in_dom_JJ_holds_
not_JJ_._x_is_empty
A19: dom JJ = {1} by FINSEQ_1:2, FINSEQ_1:def_8;
let x be set ; ::_thesis: ( x in dom JJ implies not JJ . x is empty )
assume x in dom JJ ; ::_thesis: not JJ . x is empty
then ( JJ . 1 = Jordan_block (L,(len b1)) & x = 1 ) by A19, FINSEQ_1:def_8, TARSKI:def_1;
hence not JJ . x is empty by A13, Def1; ::_thesis: verum
end;
then reconsider JJ = JJ as non-empty FinSequence_of_Jordan_block of L,K by FUNCT_1:def_9;
reconsider F9 = F as Function of V1,V2 ;
reconsider BB = block_diagonal (JJ,(0. K)) as Matrix of len b1, len b2,K by A8, MATRIXJ1:34;
set T = Mx2Tran (BB,b1,b2);
A20: block_diagonal (JJ,(0. K)) = Jordan_block (L,(len b1)) by MATRIXJ1:34;
A21: now__::_thesis:_for_y_being_set_st_y_in_dom_b1_holds_
(F9_*_b1)_._y_=_((Mx2Tran_(BB,b1,b2))_*_b1)_._y
let y be set ; ::_thesis: ( y in dom b1 implies (F9 * b1) . y = ((Mx2Tran (BB,b1,b2)) * b1) . y )
assume A22: y in dom b1 ; ::_thesis: (F9 * b1) . y = ((Mx2Tran (BB,b1,b2)) * b1) . y
reconsider j = y as Element of NAT by A22;
A23: ((Mx2Tran (BB,b1,b2)) * b1) . y = (Mx2Tran (BB,b1,b2)) . (b1 . y) by A22, FUNCT_1:13;
A24: b1 /. j = b1 . j by A22, PARTFUN1:def_6;
A25: (F9 * b1) . y = F . (b1 . y) by A22, FUNCT_1:13;
now__::_thesis:_((Mx2Tran_(BB,b1,b2))_*_b1)_._y_=_(F9_*_b1)_._y
percases ( j = len b1 or j <> len b1 ) ;
supposeA26: j = len b1 ; ::_thesis: ((Mx2Tran (BB,b1,b2)) * b1) . y = (F9 * b1) . y
hence ((Mx2Tran (BB,b1,b2)) * b1) . y = L * (b2 /. (len b1)) by A20, A22, A23, A24, Th23
.= (F9 * b1) . y by A11, A16, A15, A25, A24, A26 ;
::_thesis: verum
end;
supposeA27: j <> len b1 ; ::_thesis: ((Mx2Tran (BB,b1,b2)) * b1) . y = (F9 * b1) . y
ex z being set st FF = {z} by A10, A18, CARD_2:42;
then FF = {(len b1)} by A17, TARSKI:def_1;
then not j in FF by A27, TARSKI:def_1;
then A28: F . (b1 /. j) <> L * (b2 /. j) by A22;
((Mx2Tran (BB,b1,b2)) * b1) . y = (L * (b2 /. j)) + (b2 /. (j + 1)) by A20, A22, A23, A24, A27, Th23;
hence ((Mx2Tran (BB,b1,b2)) * b1) . y = (F9 * b1) . y by A11, A22, A25, A24, A28; ::_thesis: verum
end;
end;
end;
hence (F9 * b1) . y = ((Mx2Tran (BB,b1,b2)) * b1) . y ; ::_thesis: verum
end;
take JJ ; ::_thesis: AutMt (F,b1,b2) = block_diagonal (JJ,(0. K))
A29: rng b1 c= [#] V1 by FINSEQ_1:def_4;
dom (Mx2Tran (BB,b1,b2)) = [#] V1 by FUNCT_2:def_1;
then A30: dom ((Mx2Tran (BB,b1,b2)) * b1) = dom b1 by A29, RELAT_1:27;
dom F = [#] V1 by FUNCT_2:def_1;
then dom (F9 * b1) = dom b1 by A29, RELAT_1:27;
then F = Mx2Tran (BB,b1,b2) by A13, A30, A21, FUNCT_1:2, MATRLIN:22;
hence AutMt (F,b1,b2) = block_diagonal (JJ,(0. K)) by MATRLIN2:36; ::_thesis: verum
end;
supposeA31: n <> 0 ; ::_thesis: ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
defpred S2[ Nat] means ( $1 in FF & $1 < len b1 );
A32: ex k being Nat st S2[k]
proof
card FF <> 1 by A10, A31;
then FF <> {(len b1)} by CARD_2:42;
then consider y being set such that
A33: y in FF and
A34: y <> len b1 by A17, ZFMISC_1:35;
consider j being Element of NAT such that
A35: j = y and
A36: j in dom b1 and
F . (b1 /. j) = L * (b2 /. j) by A33;
take j ; ::_thesis: S2[j]
j <= len b1 by A36, FINSEQ_3:25;
hence S2[j] by A33, A34, A35, XXREAL_0:1; ::_thesis: verum
end;
A37: for k being Nat st S2[k] holds
k <= len b1 ;
consider m being Nat such that
A38: S2[m] and
A39: for k being Nat st S2[k] holds
k <= m from NAT_1:sch_6(A37, A32);
set b1m = b1 | m;
A40: len (b1 | m) = m by A38, FINSEQ_1:59;
set JB = Jordan_block (L,((len b1) -' m));
reconsider JJ = <*(Jordan_block (L,((len b1) -' m)))*> as FinSequence_of_Jordan_block of L,K by Th11;
set b19m = b1 /^ m;
A41: len (b1 /^ m) = (len b1) - m by A38, RFINSEQ:def_1;
set b29m = b2 /^ m;
A42: len (b2 /^ m) = (len b1) - m by A8, A38, RFINSEQ:def_1;
set b2m = b2 | m;
A43: len (b2 | m) = m by A8, A38, FINSEQ_1:59;
reconsider Rb2 = rng (b2 | m), Rb29 = rng (b2 /^ m) as linearly-independent Subset of V2 by MATRLIN2:22, MATRLIN2:23;
reconsider Rb1 = rng (b1 | m), Rb19 = rng (b1 /^ m) as linearly-independent Subset of V1 by MATRLIN2:22, MATRLIN2:23;
set Lb1 = Lin Rb1;
set Lb2 = Lin Rb2;
set Lb19 = Lin Rb19;
set Lb29 = Lin Rb29;
set FRb1 = F .: Rb1;
A44: rng (F | (Lin Rb1)) = F .: the carrier of (Lin Rb1) by RELAT_1:115;
reconsider b2m = b2 | m as OrdBasis of Lin Rb2 by MATRLIN2:22;
reconsider b1m = b1 | m as OrdBasis of Lin Rb1 by MATRLIN2:22;
A45: dom b1m = dom b2m by A40, A43, FINSEQ_3:29;
reconsider b19m = b1 /^ m as OrdBasis of Lin Rb19 by MATRLIN2:23;
reconsider b29m = b2 /^ m as OrdBasis of Lin Rb29 by MATRLIN2:23;
A46: b2 = b2m ^ b29m by RFINSEQ:8;
A47: b1 = b1m ^ b19m by RFINSEQ:8;
then A48: dom b1m c= dom b1 by FINSEQ_1:26;
A49: for i being Nat holds
( not i in dom b1m or (F | (Lin Rb1)) . (b1m /. i) = L * (b2m /. i) or ( i + 1 in dom b1m & (F | (Lin Rb1)) . (b1m /. i) = (L * (b2m /. i)) + (b2m /. (i + 1)) ) )
proof
let i be Nat; ::_thesis: ( not i in dom b1m or (F | (Lin Rb1)) . (b1m /. i) = L * (b2m /. i) or ( i + 1 in dom b1m & (F | (Lin Rb1)) . (b1m /. i) = (L * (b2m /. i)) + (b2m /. (i + 1)) ) )
assume A50: i in dom b1m ; ::_thesis: ( (F | (Lin Rb1)) . (b1m /. i) = L * (b2m /. i) or ( i + 1 in dom b1m & (F | (Lin Rb1)) . (b1m /. i) = (L * (b2m /. i)) + (b2m /. (i + 1)) ) )
A51: b1m . i = b1m /. i by A50, PARTFUN1:def_6;
set i1 = i + 1;
A52: F . (b1m /. i) = (F | (Lin Rb1)) . (b1m /. i) by FUNCT_1:49;
A53: ( b2 /. i = b2 . i & b2 . i = b2m . i ) by A9, A46, A48, A45, A50, FINSEQ_1:def_7, PARTFUN1:def_6;
A54: ( b1 /. i = b1 . i & b1 . i = b1m . i ) by A47, A48, A50, FINSEQ_1:def_7, PARTFUN1:def_6;
percases ( F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) by A11, A48, A50;
suppose F . (b1 /. i) = L * (b2 /. i) ; ::_thesis: ( (F | (Lin Rb1)) . (b1m /. i) = L * (b2m /. i) or ( i + 1 in dom b1m & (F | (Lin Rb1)) . (b1m /. i) = (L * (b2m /. i)) + (b2m /. (i + 1)) ) )
hence ( (F | (Lin Rb1)) . (b1m /. i) = L * (b2m /. i) or ( i + 1 in dom b1m & (F | (Lin Rb1)) . (b1m /. i) = (L * (b2m /. i)) + (b2m /. (i + 1)) ) ) by A45, A50, A53, A54, A51, A52, PARTFUN1:def_6, VECTSP_4:14; ::_thesis: verum
end;
supposeA55: ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ; ::_thesis: ( (F | (Lin Rb1)) . (b1m /. i) = L * (b2m /. i) or ( i + 1 in dom b1m & (F | (Lin Rb1)) . (b1m /. i) = (L * (b2m /. i)) + (b2m /. (i + 1)) ) )
reconsider rngb2 = rng b2 as Basis of V2 by MATRLIN:def_2;
A56: i + 1 <= m
proof
assume i + 1 > m ; ::_thesis: contradiction
then A57: i >= m by NAT_1:13;
i <= m by A40, A50, FINSEQ_3:25;
then A58: i = m by A57, XXREAL_0:1;
ex ii being Element of NAT st
( m = ii & ii in dom b1 & F . (b1 /. ii) = L * (b2 /. ii) ) by A38;
then F . (b1 /. i) = (L * (b2 /. i)) + (0. V2) by A58, RLVECT_1:def_4;
then A59: b2 /. (i + 1) = 0. V2 by A55, RLVECT_1:8;
A60: rngb2 is linearly-independent by VECTSP_7:def_3;
( b2 /. (i + 1) = b2 . (i + 1) & b2 . (i + 1) in rngb2 ) by A9, A55, FUNCT_1:def_3, PARTFUN1:def_6;
hence contradiction by A59, A60, VECTSP_7:2; ::_thesis: verum
end;
A61: 1 <= i + 1 by NAT_1:11;
then A62: i + 1 in dom b1m by A40, A56, FINSEQ_3:25;
then A63: b2m . (i + 1) = b2m /. (i + 1) by A45, PARTFUN1:def_6;
A64: L * (b2 /. i) = L * (b2m /. i) by A45, A50, A53, PARTFUN1:def_6, VECTSP_4:14;
( b2 /. (i + 1) = b2 . (i + 1) & b2 . (i + 1) = b2m . (i + 1) ) by A9, A46, A48, A45, A62, FINSEQ_1:def_7, PARTFUN1:def_6;
hence ( (F | (Lin Rb1)) . (b1m /. i) = L * (b2m /. i) or ( i + 1 in dom b1m & (F | (Lin Rb1)) . (b1m /. i) = (L * (b2m /. i)) + (b2m /. (i + 1)) ) ) by A40, A54, A51, A52, A55, A56, A61, A63, A64, FINSEQ_3:25, VECTSP_4:13; ::_thesis: verum
end;
end;
end;
F .: Rb1 c= the carrier of (Lin Rb2)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in F .: Rb1 or y in the carrier of (Lin Rb2) )
assume y in F .: Rb1 ; ::_thesis: y in the carrier of (Lin Rb2)
then consider x being set such that
x in dom F and
A65: x in Rb1 and
A66: F . x = y by FUNCT_1:def_6;
consider i being set such that
A67: i in dom b1m and
A68: b1m . i = x by A65, FUNCT_1:def_3;
reconsider i = i as Element of NAT by A67;
b1m /. i = b1m . i by A67, PARTFUN1:def_6;
then (F | (Lin Rb1)) . (b1m /. i) = y by A66, A68, FUNCT_1:49;
then ( y = L * (b2m /. i) or y = (L * (b2m /. i)) + (b2m /. (i + 1)) ) by A49, A67;
hence y in the carrier of (Lin Rb2) ; ::_thesis: verum
end;
then ( the carrier of (Lin (F .: Rb1)) = F .: the carrier of (Lin Rb1) & Lin (F .: Rb1) is Subspace of Lin Rb2 ) by VECTSP11:42, VECTSP_9:16;
then F .: the carrier of (Lin Rb1) c= the carrier of (Lin Rb2) by VECTSP_4:def_2;
then reconsider FL = F | (Lin Rb1) as linear-transformation of (Lin Rb1),(Lin Rb2) by A44, Lm4, FUNCT_2:6;
A69: for i being Nat holds
( not i in dom b1m or FL . (b1m /. i) = L * (b2m /. i) or ( i + 1 in dom b1m & FL . (b1m /. i) = (L * (b2m /. i)) + (b2m /. (i + 1)) ) ) by A49;
set FF1 = { i where i is Element of NAT : ( i in dom b1m & FL . (b1m /. i) = L * (b2m /. i) ) } ;
A70: FF \ {(len b1)} c= { i where i is Element of NAT : ( i in dom b1m & FL . (b1m /. i) = L * (b2m /. i) ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in FF \ {(len b1)} or x in { i where i is Element of NAT : ( i in dom b1m & FL . (b1m /. i) = L * (b2m /. i) ) } )
assume A71: x in FF \ {(len b1)} ; ::_thesis: x in { i where i is Element of NAT : ( i in dom b1m & FL . (b1m /. i) = L * (b2m /. i) ) }
A72: x <> len b1 by A71, ZFMISC_1:56;
x in FF by A71;
then consider j being Element of NAT such that
A73: j = x and
A74: j in dom b1 and
A75: F . (b1 /. j) = L * (b2 /. j) ;
j <= len b1 by A74, FINSEQ_3:25;
then j < len b1 by A72, A73, XXREAL_0:1;
then A76: j <= m by A39, A71, A73;
1 <= j by A74, FINSEQ_3:25;
then A77: j in dom b1m by A40, A76, FINSEQ_3:25;
then A78: ( b1m /. j = b1m . j & b1m . j = b1 . j ) by A47, FINSEQ_1:def_7, PARTFUN1:def_6;
A79: ( FL . (b1m /. j) = F . (b1m /. j) & b1 /. j = b1 . j ) by A48, A77, FUNCT_1:49, PARTFUN1:def_6;
( b2m . j = b2 . j & b2 /. j = b2 . j ) by A9, A46, A48, A45, A77, FINSEQ_1:def_7, PARTFUN1:def_6;
then FL . (b1m /. j) = L * (b2m /. j) by A45, A75, A77, A78, A79, PARTFUN1:def_6, VECTSP_4:14;
hence x in { i where i is Element of NAT : ( i in dom b1m & FL . (b1m /. i) = L * (b2m /. i) ) } by A73, A77; ::_thesis: verum
end;
{ i where i is Element of NAT : ( i in dom b1m & FL . (b1m /. i) = L * (b2m /. i) ) } c= FF \ {(len b1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { i where i is Element of NAT : ( i in dom b1m & FL . (b1m /. i) = L * (b2m /. i) ) } or x in FF \ {(len b1)} )
assume x in { i where i is Element of NAT : ( i in dom b1m & FL . (b1m /. i) = L * (b2m /. i) ) } ; ::_thesis: x in FF \ {(len b1)}
then consider j being Element of NAT such that
A80: x = j and
A81: j in dom b1m and
A82: FL . (b1m /. j) = L * (b2m /. j) ;
A83: ( b1m /. j = b1m . j & b1m . j = b1 . j ) by A47, A81, FINSEQ_1:def_7, PARTFUN1:def_6;
A84: ( FL . (b1m /. j) = F . (b1m /. j) & b1 /. j = b1 . j ) by A48, A81, FUNCT_1:49, PARTFUN1:def_6;
( b2m . j = b2 . j & b2 /. j = b2 . j ) by A9, A46, A48, A45, A81, FINSEQ_1:def_7, PARTFUN1:def_6;
then F . (b1 /. j) = L * (b2 /. j) by A45, A81, A82, A83, A84, PARTFUN1:def_6, VECTSP_4:14;
then A85: j in FF by A48, A81;
j <= m by A40, A81, FINSEQ_3:25;
hence x in FF \ {(len b1)} by A38, A80, A85, ZFMISC_1:56; ::_thesis: verum
end;
then { i where i is Element of NAT : ( i in dom b1m & FL . (b1m /. i) = L * (b2m /. i) ) } = FF \ {(len b1)} by A70, XBOOLE_0:def_10;
then card { i where i is Element of NAT : ( i in dom b1m & FL . (b1m /. i) = L * (b2m /. i) ) } = n by A10, A17, STIRL2_1:55;
then consider J being non-empty FinSequence_of_Jordan_block of L,K such that
A86: AutMt (FL,b1m,b2m) = block_diagonal (J,(0. K)) by A7, A40, A43, A69;
set JJJ = J ^ JJ;
A87: (len b1) - m = (len b1) -' m by A38, XREAL_1:233;
A88: now__::_thesis:_for_x_being_set_st_x_in_dom_(J_^_JJ)_holds_
not_(J_^_JJ)_._x_is_empty
let x be set ; ::_thesis: ( x in dom (J ^ JJ) implies not (J ^ JJ) . b1 is empty )
assume A89: x in dom (J ^ JJ) ; ::_thesis: not (J ^ JJ) . b1 is empty
reconsider i = x as Nat by A89;
percases ( i in dom J or ex j being Nat st
( j in dom JJ & i = (len J) + j ) ) by A89, FINSEQ_1:25;
supposeA90: i in dom J ; ::_thesis: not (J ^ JJ) . b1 is empty
then not J . i is empty by FUNCT_1:def_9;
hence not (J ^ JJ) . x is empty by A90, FINSEQ_1:def_7; ::_thesis: verum
end;
supposeA91: ex j being Nat st
( j in dom JJ & i = (len J) + j ) ; ::_thesis: not (J ^ JJ) . b1 is empty
len (Jordan_block (L,((len b1) -' m))) = (len b1) -' m by Def1;
then A92: len (Jordan_block (L,((len b1) -' m))) <> 0 by A38, A87;
consider j being Nat such that
A93: j in dom JJ and
A94: i = (len J) + j by A91;
dom JJ = Seg 1 by FINSEQ_1:38;
then j = 1 by A93, FINSEQ_1:2, TARSKI:def_1;
then (J ^ JJ) . i = JJ . 1 by A93, A94, FINSEQ_1:def_7;
hence not (J ^ JJ) . x is empty by A92, FINSEQ_1:40; ::_thesis: verum
end;
end;
end;
reconsider JB = Jordan_block (L,((len b1) -' m)) as Matrix of len b19m, len b29m,K by A8, A38, A41, A87, RFINSEQ:def_1;
A95: width JB = len b19m by A41, A42, MATRIX_1:24;
reconsider JJJ = J ^ JJ as non-empty FinSequence_of_Jordan_block of L,K by A88, FUNCT_1:def_9;
take JJJ ; ::_thesis: AutMt (F,b1,b2) = block_diagonal (JJJ,(0. K))
reconsider F9 = F as Function of V1,V2 ;
reconsider B = block_diagonal (J,(0. K)) as Matrix of len b1m, len b2m,K by A86;
A96: width B = len b1m by A40, A43, MATRIX_1:24;
A97: ( Sum (Len <*B,JB*>) = (len B) + (len JB) & Sum (Width <*B,JB*>) = (width B) + (width JB) ) by MATRIXJ1:16, MATRIXJ1:20;
len B = len b1m by A40, A43, MATRIX_1:24;
then reconsider BB = block_diagonal (<*B,JB*>,(0. K)) as Matrix of len b1, len b2,K by A8, A40, A41, A42, A97, A96, A95, MATRIX_1:24;
set T = Mx2Tran (BB,b1,b2);
A98: dom b19m = dom b29m by A41, A42, FINSEQ_3:29;
A99: now__::_thesis:_for_x_being_set_st_x_in_dom_b1_holds_
(F9_*_b1)_._x_=_((Mx2Tran_(BB,b1,b2))_*_b1)_._x
let x be set ; ::_thesis: ( x in dom b1 implies (F9 * b1) . x = ((Mx2Tran (BB,b1,b2)) * b1) . x )
assume A100: x in dom b1 ; ::_thesis: (F9 * b1) . x = ((Mx2Tran (BB,b1,b2)) * b1) . x
reconsider I = x as Element of NAT by A100;
A101: ((Mx2Tran (BB,b1,b2)) * b1) . x = (Mx2Tran (BB,b1,b2)) . (b1 . I) by A100, FUNCT_1:13;
A102: b1 . I = b1 /. I by A100, PARTFUN1:def_6;
A103: (F9 * b1) . x = F . (b1 . I) by A100, FUNCT_1:13;
now__::_thesis:_((Mx2Tran_(BB,b1,b2))_*_b1)_._x_=_(F9_*_b1)_._x
percases ( I in dom b1m or ex j being Nat st
( j in dom b19m & I = (len b1m) + j ) ) by A47, A100, FINSEQ_1:25;
supposeA104: I in dom b1m ; ::_thesis: ((Mx2Tran (BB,b1,b2)) * b1) . x = (F9 * b1) . x
then A105: ( b1m /. I = b1m . I & b1 . I = b1m . I ) by A47, FINSEQ_1:def_7, PARTFUN1:def_6;
A106: FL . (b1m /. I) = F . (b1m /. I) by FUNCT_1:49;
thus ((Mx2Tran (BB,b1,b2)) * b1) . x = (Mx2Tran (B,b1m,b2m)) . (b1m /. I) by A40, A43, A41, A42, A47, A46, A96, A95, A101, A102, A104, Th19
.= FL . (b1m /. I) by A86, MATRLIN2:34
.= (F9 * b1) . x by A100, A106, A105, FUNCT_1:13 ; ::_thesis: verum
end;
suppose ex j being Nat st
( j in dom b19m & I = (len b1m) + j ) ; ::_thesis: (F9 * b1) . x = ((Mx2Tran (BB,b1,b2)) * b1) . x
then consider j being Nat such that
A107: j in dom b19m and
A108: I = (len b1m) + j ;
now__::_thesis:_(F9_*_b1)_._x_=_(Mx2Tran_(JB,b19m,b29m))_._(b19m_/._j)
percases ( j = len b19m or j <> len b19m ) ;
supposeA109: j = len b19m ; ::_thesis: (F9 * b1) . x = (Mx2Tran (JB,b19m,b29m)) . (b19m /. j)
A110: ( b2 . I = b29m . j & b2 /. I = b2 . I ) by A9, A40, A43, A46, A98, A100, A107, A108, FINSEQ_1:def_7, PARTFUN1:def_6;
thus (F9 * b1) . x = L * (b2 /. I) by A11, A16, A15, A40, A41, A103, A102, A108, A109
.= L * (b29m /. j) by A98, A107, A110, PARTFUN1:def_6, VECTSP_4:14
.= (Mx2Tran (JB,b19m,b29m)) . (b19m /. j) by A107, A109, Th23 ; ::_thesis: verum
end;
supposeA111: j <> len b19m ; ::_thesis: (F9 * b1) . x = (Mx2Tran (JB,b19m,b29m)) . (b19m /. j)
A112: (F9 * b1) . x = (L * (b2 /. I)) + (b2 /. (I + 1))
proof
assume (F9 * b1) . x <> (L * (b2 /. I)) + (b2 /. (I + 1)) ; ::_thesis: contradiction
then F . (b1 /. I) = L * (b2 /. I) by A11, A100, A103, A102;
then A113: I in FF by A100;
( I <> len b1 & I <= len b1 ) by A40, A41, A100, A108, A111, FINSEQ_3:25;
then I < len b1 by XXREAL_0:1;
then (len b1m) + j <= (len b1m) + 0 by A39, A40, A108, A113;
then j <= 0 by XREAL_1:6;
hence contradiction by A107, FINSEQ_3:25; ::_thesis: verum
end;
j <= len b19m by A107, FINSEQ_3:25;
then j < len b19m by A111, XXREAL_0:1;
then ( 1 <= j + 1 & j + 1 <= len b19m ) by NAT_1:11, NAT_1:13;
then A114: j + 1 in dom b19m by FINSEQ_3:25;
then ( b29m /. (j + 1) = b29m . (j + 1) & b2 . ((len b1m) + (j + 1)) = b29m . (j + 1) ) by A40, A43, A46, A98, FINSEQ_1:def_7, PARTFUN1:def_6;
then A115: b29m /. (j + 1) = b2 /. (I + 1) by A9, A47, A108, A114, FINSEQ_1:28, PARTFUN1:def_6;
( b2 . I = b29m . j & b2 /. I = b2 . I ) by A9, A40, A43, A46, A98, A100, A107, A108, FINSEQ_1:def_7, PARTFUN1:def_6;
then L * (b29m /. j) = L * (b2 /. I) by A98, A107, PARTFUN1:def_6, VECTSP_4:14;
then (L * (b2 /. I)) + (b2 /. (I + 1)) = (L * (b29m /. j)) + (b29m /. (j + 1)) by A115, VECTSP_4:13
.= (Mx2Tran (JB,b19m,b29m)) . (b19m /. j) by A107, A111, Th23 ;
hence (F9 * b1) . x = (Mx2Tran (JB,b19m,b29m)) . (b19m /. j) by A112; ::_thesis: verum
end;
end;
end;
hence (F9 * b1) . x = ((Mx2Tran (BB,b1,b2)) * b1) . x by A40, A43, A41, A42, A47, A46, A96, A95, A101, A102, A107, A108, Th19; ::_thesis: verum
end;
end;
end;
hence (F9 * b1) . x = ((Mx2Tran (BB,b1,b2)) * b1) . x ; ::_thesis: verum
end;
A116: rng b1 c= [#] V1 by FINSEQ_1:def_4;
dom (Mx2Tran (BB,b1,b2)) = [#] V1 by FUNCT_2:def_1;
then A117: dom ((Mx2Tran (BB,b1,b2)) * b1) = dom b1 by A116, RELAT_1:27;
dom F = [#] V1 by FUNCT_2:def_1;
then dom (F9 * b1) = dom b1 by A116, RELAT_1:27;
then ( block_diagonal (JJJ,(0. K)) = block_diagonal (<*B,JB*>,(0. K)) & F = Mx2Tran (BB,b1,b2) ) by A13, A117, A99, FUNCT_1:2, MATRIXJ1:35, MATRLIN:22;
hence AutMt (F,b1,b2) = block_diagonal (JJJ,(0. K)) by MATRLIN2:36; ::_thesis: verum
end;
end;
end;
let V1, V2 be finite-dimensional VectSp of K; ::_thesis: for b1 being OrdBasis of V1
for b2 being OrdBasis of V2
for L being Element of K st len b1 = len b2 holds
for F being linear-transformation of V1,V2 st ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
let b1 be OrdBasis of V1; ::_thesis: for b2 being OrdBasis of V2
for L being Element of K st len b1 = len b2 holds
for F being linear-transformation of V1,V2 st ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
let b2 be OrdBasis of V2; ::_thesis: for L being Element of K st len b1 = len b2 holds
for F being linear-transformation of V1,V2 st ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
let L be Element of K; ::_thesis: ( len b1 = len b2 implies for F being linear-transformation of V1,V2 st ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K)) )
assume A118: len b1 = len b2 ; ::_thesis: for F being linear-transformation of V1,V2 st ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) holds
ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
let F be linear-transformation of V1,V2; ::_thesis: ( ( for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ) implies ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K)) )
assume A119: for i being Nat holds
( not i in dom b1 or F . (b1 /. i) = L * (b2 /. i) or ( i + 1 in dom b1 & F . (b1 /. i) = (L * (b2 /. i)) + (b2 /. (i + 1)) ) ) ; ::_thesis: ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K))
set FF = { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } ;
{ i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } c= dom b1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } or x in dom b1 )
assume x in { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } ; ::_thesis: x in dom b1
then ex i being Element of NAT st
( x = i & i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) ;
hence x in dom b1 ; ::_thesis: verum
end;
then reconsider FF = { i where i is Element of NAT : ( i in dom b1 & F . (b1 /. i) = L * (b2 /. i) ) } as finite set ;
for n being Nat holds S1[n] from NAT_1:sch_2(A1, A6);
then S1[ card FF] ;
hence ex J being non-empty FinSequence_of_Jordan_block of L,K st AutMt (F,b1,b2) = block_diagonal (J,(0. K)) by A118, A119; ::_thesis: verum
end;
theorem Th29: :: MATRIXJ2:29
for K being Field
for V1 being finite-dimensional VectSp of K
for F being nilpotent linear-transformation of V1,V1 ex J being non-empty FinSequence_of_Jordan_block of 0. K,K ex b1 being OrdBasis of V1 st AutMt (F,b1,b1) = block_diagonal (J,(0. K))
proof
let K be Field; ::_thesis: for V1 being finite-dimensional VectSp of K
for F being nilpotent linear-transformation of V1,V1 ex J being non-empty FinSequence_of_Jordan_block of 0. K,K ex b1 being OrdBasis of V1 st AutMt (F,b1,b1) = block_diagonal (J,(0. K))
defpred S1[ Nat] means for V1 being finite-dimensional VectSp of K
for F being nilpotent linear-transformation of V1,V1 st deg F = $1 holds
ex J being FinSequence_of_Jordan_block of 0. K,K ex b1 being OrdBasis of V1 st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for i being Nat st i in dom J holds
(Len J) . i <> 0 ) );
let V1 be finite-dimensional VectSp of K; ::_thesis: for F being nilpotent linear-transformation of V1,V1 ex J being non-empty FinSequence_of_Jordan_block of 0. K,K ex b1 being OrdBasis of V1 st AutMt (F,b1,b1) = block_diagonal (J,(0. K))
let F be nilpotent linear-transformation of V1,V1; ::_thesis: ex J being non-empty FinSequence_of_Jordan_block of 0. K,K ex b1 being OrdBasis of V1 st AutMt (F,b1,b1) = block_diagonal (J,(0. K))
A1: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A2: S1[n] ; ::_thesis: S1[n + 1]
let V1 be finite-dimensional VectSp of K; ::_thesis: for F being nilpotent linear-transformation of V1,V1 st deg F = n + 1 holds
ex J being FinSequence_of_Jordan_block of 0. K,K ex b1 being OrdBasis of V1 st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for i being Nat st i in dom J holds
(Len J) . i <> 0 ) )
set n1 = n + 1;
let F be nilpotent linear-transformation of V1,V1; ::_thesis: ( deg F = n + 1 implies ex J being FinSequence_of_Jordan_block of 0. K,K ex b1 being OrdBasis of V1 st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for i being Nat st i in dom J holds
(Len J) . i <> 0 ) ) )
assume A3: deg F = n + 1 ; ::_thesis: ex J being FinSequence_of_Jordan_block of 0. K,K ex b1 being OrdBasis of V1 st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for i being Nat st i in dom J holds
(Len J) . i <> 0 ) )
set BAS = the Basis of V1;
A4: the Basis of V1 is linearly-independent by VECTSP_7:def_3;
A5: Lin the Basis of V1 = VectSpStr(# the carrier of V1, the addF of V1, the ZeroF of V1, the lmult of V1 #) by VECTSP_7:def_3;
set IM = im (F |^ 1);
reconsider FI = F | (im (F |^ 1)) as linear-transformation of (im (F |^ 1)),(im (F |^ 1)) by VECTSP11:32;
reconsider FI = FI as nilpotent linear-transformation of (im (F |^ 1)),(im (F |^ 1)) by Th17;
(deg FI) + 1 = n + 1 by A3, Th18, NAT_1:11;
then consider J being FinSequence_of_Jordan_block of 0. K,K, b1 being OrdBasis of im (F |^ 1) such that
A6: AutMt (FI,b1,b1) = block_diagonal (J,(0. K)) and
A7: for i being Nat st i in dom J holds
(Len J) . i <> 0 by A2;
A8: len b1 = len (AutMt (FI,b1,b1)) by MATRIX_1:def_2
.= Sum (Len J) by A6, MATRIXJ1:def_5 ;
then A9: dom b1 = Seg (Sum (Len J)) by FINSEQ_1:def_3;
set L = len J;
set LJ = Len J;
set S = Sum (Len J);
defpred S2[ Nat, Nat] means ( $2 in dom (Len J) & $2 <= $1 & Sum ((Len J) | ($2 -' 1)) <= $1 -' $2 );
defpred S3[ set , set ] means for i, k being Nat st i = $1 & k = $2 holds
( S2[i,k] & i -' k <= Sum ((Len J) | k) & ( for n being Nat st S2[i,n] holds
n <= k ) );
A10: for x being set st x in Seg ((Sum (Len J)) + (len J)) holds
ex y being set st
( y in NAT & S3[x,y] )
proof
let x be set ; ::_thesis: ( x in Seg ((Sum (Len J)) + (len J)) implies ex y being set st
( y in NAT & S3[x,y] ) )
assume A11: x in Seg ((Sum (Len J)) + (len J)) ; ::_thesis: ex y being set st
( y in NAT & S3[x,y] )
reconsider i = x as Nat by A11;
len J <> 0
proof
assume A12: len J = 0 ; ::_thesis: contradiction
then Len J = <*> NAT ;
hence contradiction by A11, A12, RVSUM_1:72; ::_thesis: verum
end;
then A13: 1 <= len J by NAT_1:14;
1 -' 1 = 0 by XREAL_1:232;
then A14: Sum ((Len J) | (1 -' 1)) = 0 by RVSUM_1:72;
defpred S4[ Nat] means ( $1 in dom (Len J) & $1 <= i & Sum ((Len J) | ($1 -' 1)) <= i -' $1 );
A15: for k being Nat st S4[k] holds
k <= len J
proof
let k be Nat; ::_thesis: ( S4[k] implies k <= len J )
assume S4[k] ; ::_thesis: k <= len J
then k <= len (Len J) by FINSEQ_3:25;
hence k <= len J by CARD_1:def_7; ::_thesis: verum
end;
len (Len J) = len J by CARD_1:def_7;
then A16: ( 0 <= i -' 1 & 1 in dom (Len J) ) by A13, FINSEQ_3:25;
1 <= i by A11, FINSEQ_1:1;
then A17: ex k being Nat st S4[k] by A14, A16;
consider k being Nat such that
A18: S4[k] and
A19: for n being Nat st S4[n] holds
n <= k from NAT_1:sch_6(A15, A17);
A20: i -' k <= Sum ((Len J) | k)
proof
assume A21: i -' k > Sum ((Len J) | k) ; ::_thesis: contradiction
then i -' k >= (Sum ((Len J) | k)) + 1 by NAT_1:13;
then A22: (i -' k) - 1 >= ((Sum ((Len J) | k)) + 1) - 1 by XREAL_1:9;
A23: i -' k = i - k by A18, XREAL_1:233;
A24: k + 1 <= len (Len J)
proof
assume k + 1 > len (Len J) ; ::_thesis: contradiction
then A25: k >= len (Len J) by NAT_1:13;
then i - k > Sum (Len J) by A21, A23, FINSEQ_1:58;
then A26: (i - k) + k > (Sum (Len J)) + k by XREAL_1:6;
len (Len J) = len J by CARD_1:def_7;
then (Sum (Len J)) + k >= (Sum (Len J)) + (len J) by A25, XREAL_1:6;
then i > (Sum (Len J)) + (len J) by A26, XXREAL_0:2;
hence contradiction by A11, FINSEQ_1:1; ::_thesis: verum
end;
1 <= k + 1 by NAT_1:14;
then A27: k + 1 in dom (Len J) by A24, FINSEQ_3:25;
i -' k >= 1 by A21, NAT_1:14;
then A28: (i - k) + k >= 1 + k by A23, XREAL_1:6;
then i -' (k + 1) = i - (k + 1) by XREAL_1:233;
then Sum ((Len J) | ((k + 1) -' 1)) <= i -' (k + 1) by A22, A23, NAT_D:34;
then k + 1 <= k by A19, A28, A27;
hence contradiction by NAT_1:13; ::_thesis: verum
end;
take k ; ::_thesis: ( k in NAT & S3[x,k] )
thus k in NAT by ORDINAL1:def_12; ::_thesis: S3[x,k]
let i9, k9 be Nat; ::_thesis: ( i9 = x & k9 = k implies ( S2[i9,k9] & i9 -' k9 <= Sum ((Len J) | k9) & ( for n being Nat st S2[i9,n] holds
n <= k9 ) ) )
assume ( i9 = x & k9 = k ) ; ::_thesis: ( S2[i9,k9] & i9 -' k9 <= Sum ((Len J) | k9) & ( for n being Nat st S2[i9,n] holds
n <= k9 ) )
hence ( S2[i9,k9] & i9 -' k9 <= Sum ((Len J) | k9) & ( for n being Nat st S2[i9,n] holds
n <= k9 ) ) by A18, A19, A20; ::_thesis: verum
end;
consider r being Function of (Seg ((Sum (Len J)) + (len J))),NAT such that
A29: for x being set st x in Seg ((Sum (Len J)) + (len J)) holds
S3[x,r . x] from FUNCT_2:sch_1(A10);
defpred S4[ set , set ] means for i, k being Nat st i = $1 & k = r . i holds
( ( i -' k = Sum ((Len J) | (k -' 1)) implies ( F . $2 = b1 . ((i -' k) + 1) & (i -' k) + 1 in dom b1 ) ) & ( i -' k <> Sum ((Len J) | (k -' 1)) implies ( $2 = b1 . (i -' k) & i -' k in dom b1 & min ((Len J),(i -' k)) = k & ( i -' k < Sum ((Len J) | k) implies ( F . $2 = b1 . ((i -' k) + 1) & (i -' k) + 1 in dom b1 ) ) & ( i -' k = Sum ((Len J) | k) implies F . $2 = 0. V1 ) ) ) );
A30: dom r = Seg ((Sum (Len J)) + (len J)) by FUNCT_2:def_1;
A31: FI = Mx2Tran ((AutMt (FI,b1,b1)),b1,b1) by MATRLIN2:34;
A32: for x being set st x in Seg ((Sum (Len J)) + (len J)) holds
ex y being set st
( y in the carrier of V1 & S4[x,y] )
proof
let x be set ; ::_thesis: ( x in Seg ((Sum (Len J)) + (len J)) implies ex y being set st
( y in the carrier of V1 & S4[x,y] ) )
assume A33: x in Seg ((Sum (Len J)) + (len J)) ; ::_thesis: ex y being set st
( y in the carrier of V1 & S4[x,y] )
reconsider i = x as Nat by A33;
r . i = r /. i by A30, A33, PARTFUN1:def_6;
then reconsider k = r . i as Element of NAT ;
A34: i -' k <= Sum ((Len J) | k) by A29, A33;
A35: S2[i,k] by A29, A33;
then A36: (Len J) . k = len (J . k) by MATRIXJ1:def_3;
k <= len (Len J) by A35, FINSEQ_3:25;
then A37: Sum ((Len J) | k) <= Sum ((Len J) | (len (Len J))) by POLYNOM3:18;
1 <= k by A35, FINSEQ_3:25;
then A38: k -' 1 = k - 1 by XREAL_1:233;
then k = (k -' 1) + 1 ;
then (Len J) | k = ((Len J) | (k -' 1)) ^ <*((Len J) . k)*> by A35, FINSEQ_5:10;
then A39: ( dom (Len J) = dom J & Sum ((Len J) | k) = (Sum ((Len J) | (k -' 1))) + (len (J . k)) ) by A36, MATRIXJ1:def_3, RVSUM_1:74;
A40: (Len J) | (len (Len J)) = Len J by FINSEQ_1:58;
percases ( i -' k = Sum ((Len J) | (k -' 1)) or i -' k <> Sum ((Len J) | (k -' 1)) ) ;
supposeA41: i -' k = Sum ((Len J) | (k -' 1)) ; ::_thesis: ex y being set st
( y in the carrier of V1 & S4[x,y] )
( b1 /. ((i -' k) + 1) in im (F |^ 1) & b1 /. ((i -' k) + 1) is Element of V1 ) by STRUCT_0:def_5, VECTSP_4:10;
then consider y being Element of V1 such that
A42: (F |^ 1) . y = b1 /. ((i -' k) + 1) by RANKNULL:13;
take y ; ::_thesis: ( y in the carrier of V1 & S4[x,y] )
thus y in the carrier of V1 ; ::_thesis: S4[x,y]
i -' k <> Sum ((Len J) | k) by A7, A35, A36, A39, A41;
then i -' k < Sum ((Len J) | k) by A34, XXREAL_0:1;
then (i -' k) + 1 <= Sum ((Len J) | k) by NAT_1:13;
then A43: ( 1 <= (i -' k) + 1 & (i -' k) + 1 <= Sum (Len J) ) by A37, A40, NAT_1:11, XXREAL_0:2;
then (i -' k) + 1 in dom b1 by A9;
then b1 /. ((i -' k) + 1) = b1 . ((i -' k) + 1) by PARTFUN1:def_6;
hence S4[x,y] by A9, A41, A43, A42, VECTSP11:19; ::_thesis: verum
end;
supposeA44: i -' k <> Sum ((Len J) | (k -' 1)) ; ::_thesis: ex y being set st
( y in the carrier of V1 & S4[x,y] )
take y = b1 /. (i -' k); ::_thesis: ( y in the carrier of V1 & S4[x,y] )
y in im (F |^ 1) by STRUCT_0:def_5;
then y in V1 by VECTSP_4:9;
hence y in the carrier of V1 by STRUCT_0:def_5; ::_thesis: S4[x,y]
i -' k > Sum ((Len J) | (k -' 1)) by A35, A44, XXREAL_0:1;
then A45: 1 <= i -' k by NAT_1:14;
i -' k <= Sum (Len J) by A34, A37, A40, XXREAL_0:2;
then A46: i -' k in dom b1 by A9, A45;
i -' k <= Sum ((Len J) | k) by A29, A33;
then A47: min ((Len J),(i -' k)) <= k by A9, A46, MATRIXJ1:def_1;
A48: min ((Len J),(i -' k)) = k
proof
assume min ((Len J),(i -' k)) <> k ; ::_thesis: contradiction
then min ((Len J),(i -' k)) < (k -' 1) + 1 by A38, A47, XXREAL_0:1;
then min ((Len J),(i -' k)) <= k -' 1 by NAT_1:13;
then A49: Sum ((Len J) | (min ((Len J),(i -' k)))) <= Sum ((Len J) | (k -' 1)) by POLYNOM3:18;
i -' k <= Sum ((Len J) | (min ((Len J),(i -' k)))) by A9, A46, MATRIXJ1:def_1;
then i -' k <= Sum ((Len J) | (k -' 1)) by A49, XXREAL_0:2;
hence contradiction by A35, A44, XXREAL_0:1; ::_thesis: verum
end;
A50: Len (J | k) = (Len J) | k by MATRIXJ1:17;
A51: now__::_thesis:_(_i_-'_k_=_Sum_((Len_J)_|_k)_implies_F_._y_=_0._V1_)
assume A52: i -' k = Sum ((Len J) | k) ; ::_thesis: F . y = 0. V1
F . (b1 /. (i -' k)) = FI . (b1 /. (i -' k)) by FUNCT_1:49
.= (0. K) * (b1 /. (i -' k)) by A6, A31, A46, A48, A50, A52, Th24
.= 0. (im (F |^ 1)) by VECTSP_1:14
.= 0. V1 by VECTSP_4:11 ;
hence F . y = 0. V1 ; ::_thesis: verum
end;
A53: now__::_thesis:_(_i_-'_k_<_Sum_((Len_J)_|_k)_implies_(_F_._y_=_b1_._((i_-'_k)_+_1)_&_(i_-'_k)_+_1_in_dom_b1_)_)
assume A54: i -' k < Sum ((Len J) | k) ; ::_thesis: ( F . y = b1 . ((i -' k) + 1) & (i -' k) + 1 in dom b1 )
then (i -' k) + 1 <= Sum ((Len J) | k) by NAT_1:13;
then A55: ( 1 <= (i -' k) + 1 & (i -' k) + 1 <= Sum (Len J) ) by A37, A40, NAT_1:11, XXREAL_0:2;
then A56: (i -' k) + 1 in dom b1 by A9;
F . (b1 /. (i -' k)) = FI . (b1 /. (i -' k)) by FUNCT_1:49
.= ((0. K) * (b1 /. (i -' k))) + (b1 /. ((i -' k) + 1)) by A6, A31, A46, A48, A50, A54, Th24
.= (0. (im (F |^ 1))) + (b1 /. ((i -' k) + 1)) by VECTSP_1:14
.= b1 /. ((i -' k) + 1) by RLVECT_1:def_4
.= b1 . ((i -' k) + 1) by A56, PARTFUN1:def_6 ;
hence ( F . y = b1 . ((i -' k) + 1) & (i -' k) + 1 in dom b1 ) by A9, A55; ::_thesis: verum
end;
let i9, k9 be Nat; ::_thesis: ( i9 = x & k9 = r . i9 implies ( ( i9 -' k9 = Sum ((Len J) | (k9 -' 1)) implies ( F . y = b1 . ((i9 -' k9) + 1) & (i9 -' k9) + 1 in dom b1 ) ) & ( i9 -' k9 <> Sum ((Len J) | (k9 -' 1)) implies ( y = b1 . (i9 -' k9) & i9 -' k9 in dom b1 & min ((Len J),(i9 -' k9)) = k9 & ( i9 -' k9 < Sum ((Len J) | k9) implies ( F . y = b1 . ((i9 -' k9) + 1) & (i9 -' k9) + 1 in dom b1 ) ) & ( i9 -' k9 = Sum ((Len J) | k9) implies F . y = 0. V1 ) ) ) ) )
assume ( x = i9 & k9 = r . i9 ) ; ::_thesis: ( ( i9 -' k9 = Sum ((Len J) | (k9 -' 1)) implies ( F . y = b1 . ((i9 -' k9) + 1) & (i9 -' k9) + 1 in dom b1 ) ) & ( i9 -' k9 <> Sum ((Len J) | (k9 -' 1)) implies ( y = b1 . (i9 -' k9) & i9 -' k9 in dom b1 & min ((Len J),(i9 -' k9)) = k9 & ( i9 -' k9 < Sum ((Len J) | k9) implies ( F . y = b1 . ((i9 -' k9) + 1) & (i9 -' k9) + 1 in dom b1 ) ) & ( i9 -' k9 = Sum ((Len J) | k9) implies F . y = 0. V1 ) ) ) )
hence ( ( i9 -' k9 = Sum ((Len J) | (k9 -' 1)) implies ( F . y = b1 . ((i9 -' k9) + 1) & (i9 -' k9) + 1 in dom b1 ) ) & ( i9 -' k9 <> Sum ((Len J) | (k9 -' 1)) implies ( y = b1 . (i9 -' k9) & i9 -' k9 in dom b1 & min ((Len J),(i9 -' k9)) = k9 & ( i9 -' k9 < Sum ((Len J) | k9) implies ( F . y = b1 . ((i9 -' k9) + 1) & (i9 -' k9) + 1 in dom b1 ) ) & ( i9 -' k9 = Sum ((Len J) | k9) implies F . y = 0. V1 ) ) ) ) by A44, A46, A48, A53, A51, PARTFUN1:def_6; ::_thesis: verum
end;
end;
end;
consider B being Function of (Seg ((Sum (Len J)) + (len J))), the carrier of V1 such that
A57: for x being set st x in Seg ((Sum (Len J)) + (len J)) holds
S4[x,B . x] from FUNCT_2:sch_1(A32);
A58: rng B c= the carrier of V1 by RELAT_1:def_19;
A59: dom B = Seg ((Sum (Len J)) + (len J)) by FUNCT_2:def_1;
then reconsider B = B as FinSequence by FINSEQ_1:def_2;
reconsider B = B as FinSequence of V1 by A58, FINSEQ_1:def_4;
reconsider RNG = rng B as Subset of V1 by FINSEQ_1:def_4;
now__::_thesis:_for_l_being_Linear_Combination_of_RNG_st_Sum_l_=_0._V1_holds_
Carrier_l_=_{}
rng b1 is Basis of im (F |^ 1) by MATRLIN:def_2;
then rng b1 is linearly-independent Subset of (im (F |^ 1)) by VECTSP_7:def_3;
then reconsider rngb1 = rng b1 as linearly-independent Subset of V1 by VECTSP_9:11;
set RB = { v1 where v1 is Vector of V1 : ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k <> Sum ((Len J) | (k -' 1)) & i -' k = Sum ((Len J) | k) ) } ;
set RA = { v1 where v1 is Vector of V1 : ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k < Sum ((Len J) | k) ) } ;
A60: { v1 where v1 is Vector of V1 : ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k < Sum ((Len J) | k) ) } c= the carrier of V1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { v1 where v1 is Vector of V1 : ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k < Sum ((Len J) | k) ) } or x in the carrier of V1 )
assume x in { v1 where v1 is Vector of V1 : ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k < Sum ((Len J) | k) ) } ; ::_thesis: x in the carrier of V1
then ex v1 being Vector of V1 st
( x = v1 & ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k < Sum ((Len J) | k) ) ) ;
hence x in the carrier of V1 ; ::_thesis: verum
end;
{ v1 where v1 is Vector of V1 : ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k <> Sum ((Len J) | (k -' 1)) & i -' k = Sum ((Len J) | k) ) } c= the carrier of V1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { v1 where v1 is Vector of V1 : ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k <> Sum ((Len J) | (k -' 1)) & i -' k = Sum ((Len J) | k) ) } or x in the carrier of V1 )
assume x in { v1 where v1 is Vector of V1 : ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k <> Sum ((Len J) | (k -' 1)) & i -' k = Sum ((Len J) | k) ) } ; ::_thesis: x in the carrier of V1
then ex v1 being Vector of V1 st
( x = v1 & ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k <> Sum ((Len J) | (k -' 1)) & i -' k = Sum ((Len J) | k) ) ) ;
hence x in the carrier of V1 ; ::_thesis: verum
end;
then reconsider RA = { v1 where v1 is Vector of V1 : ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k < Sum ((Len J) | k) ) } , RB = { v1 where v1 is Vector of V1 : ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k <> Sum ((Len J) | (k -' 1)) & i -' k = Sum ((Len J) | k) ) } as Subset of V1 by A60;
let l be Linear_Combination of RNG; ::_thesis: ( Sum l = 0. V1 implies Carrier l = {} )
assume A61: Sum l = 0. V1 ; ::_thesis: Carrier l = {}
A62: F | RA is one-to-one
proof
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom (F | RA) or not x2 in dom (F | RA) or not (F | RA) . x1 = (F | RA) . x2 or x1 = x2 )
assume that
A63: x1 in dom (F | RA) and
A64: x2 in dom (F | RA) and
A65: (F | RA) . x1 = (F | RA) . x2 ; ::_thesis: x1 = x2
A66: ( (F | RA) . x1 = F . x1 & (F | RA) . x2 = F . x2 ) by A63, A64, FUNCT_1:47;
A67: dom (F | RA) = (dom F) /\ RA by RELAT_1:61;
then x1 in RA by A63, XBOOLE_0:def_4;
then consider v1 being Vector of V1 such that
A68: x1 = v1 and
A69: ex i1, k1 being Nat st
( i1 in Seg ((len J) + (Sum (Len J))) & k1 = r . i1 & v1 = B . i1 & i1 -' k1 < Sum ((Len J) | k1) ) ;
consider i1, k1 being Nat such that
A70: ( i1 in Seg ((len J) + (Sum (Len J))) & k1 = r . i1 ) and
A71: v1 = B . i1 and
A72: i1 -' k1 < Sum ((Len J) | k1) by A69;
k1 <= i1 by A29, A70;
then A73: i1 -' k1 = i1 - k1 by XREAL_1:233;
A74: k1 in dom (Len J) by A29, A70;
then 1 <= k1 by FINSEQ_3:25;
then A75: k1 -' 1 = k1 - 1 by XREAL_1:233;
then (k1 -' 1) + 1 <= len (Len J) by A74, FINSEQ_3:25;
then A76: k1 -' 1 <= len (Len J) by NAT_1:13;
A77: b1 is one-to-one by MATRLIN:def_2;
A78: dom (Len J) = dom J by MATRIXJ1:def_3;
then A79: ( k1 -' 1 in dom (Len J) implies (Len J) . (k1 -' 1) <> 0 ) by A7;
x2 in RA by A64, A67, XBOOLE_0:def_4;
then consider v2 being Vector of V1 such that
A80: x2 = v2 and
A81: ex i2, k2 being Nat st
( i2 in Seg ((len J) + (Sum (Len J))) & k2 = r . i2 & v2 = B . i2 & i2 -' k2 < Sum ((Len J) | k2) ) ;
consider i2, k2 being Nat such that
A82: ( i2 in Seg ((len J) + (Sum (Len J))) & k2 = r . i2 ) and
A83: v2 = B . i2 and
A84: i2 -' k2 < Sum ((Len J) | k2) by A81;
A85: k2 <= i2 by A29, A82;
then A86: i2 -' k2 = i2 - k2 by XREAL_1:233;
A87: k2 in dom (Len J) by A29, A82;
then 1 <= k2 by FINSEQ_3:25;
then A88: k2 -' 1 = k2 - 1 by XREAL_1:233;
then (k2 -' 1) + 1 <= len (Len J) by A87, FINSEQ_3:25;
then A89: k2 -' 1 <= len (Len J) by NAT_1:13;
A90: ( k2 -' 1 in dom (Len J) implies (Len J) . (k2 -' 1) <> 0 ) by A7, A78;
percases ( ( i1 -' k1 = Sum ((Len J) | (k1 -' 1)) & i2 -' k2 = Sum ((Len J) | (k2 -' 1)) ) or ( i1 -' k1 = Sum ((Len J) | (k1 -' 1)) & i2 -' k2 <> Sum ((Len J) | (k2 -' 1)) ) or ( i1 -' k1 <> Sum ((Len J) | (k1 -' 1)) & i2 -' k2 = Sum ((Len J) | (k2 -' 1)) ) or ( i1 -' k1 <> Sum ((Len J) | (k1 -' 1)) & i2 -' k2 <> Sum ((Len J) | (k2 -' 1)) ) ) ;
supposeA91: ( i1 -' k1 = Sum ((Len J) | (k1 -' 1)) & i2 -' k2 = Sum ((Len J) | (k2 -' 1)) ) ; ::_thesis: x1 = x2
then A92: ( F . v2 = b1 . ((i2 -' k2) + 1) & (i2 -' k2) + 1 in dom b1 ) by A57, A82, A83;
( F . v1 = b1 . ((i1 -' k1) + 1) & (i1 -' k1) + 1 in dom b1 ) by A57, A70, A71, A91;
then (i1 -' k1) + 1 = (i2 -' k2) + 1 by A65, A66, A68, A80, A77, A92, FUNCT_1:def_4;
then k1 -' 1 = k2 -' 1 by A76, A89, A79, A90, A91, MATRIXJ1:11;
then i1 - k1 = i2 - k1 by A85, A73, A75, A88, A91, XREAL_1:233;
hence x1 = x2 by A68, A71, A80, A83; ::_thesis: verum
end;
supposeA93: ( i1 -' k1 = Sum ((Len J) | (k1 -' 1)) & i2 -' k2 <> Sum ((Len J) | (k2 -' 1)) ) ; ::_thesis: x1 = x2
then A94: min ((Len J),(i2 -' k2)) = k2 by A57, A82;
A95: ( F . v1 = b1 . ((i1 -' k1) + 1) & (i1 -' k1) + 1 in dom b1 ) by A57, A70, A71, A93;
( F . v2 = b1 . ((i2 -' k2) + 1) & (i2 -' k2) + 1 in dom b1 ) by A57, A82, A83, A84, A93;
then A96: (i1 -' k1) + 1 = (i2 -' k2) + 1 by A65, A66, A68, A80, A77, A95, FUNCT_1:def_4;
k1 -' 1 <> 0
proof
assume k1 -' 1 = 0 ; ::_thesis: contradiction
then (Len J) | (k1 -' 1) = <*> REAL ;
hence contradiction by A29, A82, A93, A96, RVSUM_1:72; ::_thesis: verum
end;
then k1 -' 1 >= 1 by NAT_1:14;
then A97: k1 -' 1 in dom (Len J) by A76, FINSEQ_3:25;
then (Len J) . (k1 -' 1) <> 0 by A7, A78;
hence x1 = x2 by A84, A93, A94, A96, A97, MATRIXJ1:6; ::_thesis: verum
end;
supposeA98: ( i1 -' k1 <> Sum ((Len J) | (k1 -' 1)) & i2 -' k2 = Sum ((Len J) | (k2 -' 1)) ) ; ::_thesis: x1 = x2
then A99: min ((Len J),(i1 -' k1)) = k1 by A57, A70;
A100: ( F . v2 = b1 . ((i2 -' k2) + 1) & (i2 -' k2) + 1 in dom b1 ) by A57, A82, A83, A98;
( F . v1 = b1 . ((i1 -' k1) + 1) & (i1 -' k1) + 1 in dom b1 ) by A57, A70, A71, A72, A98;
then A101: (i1 -' k1) + 1 = (i2 -' k2) + 1 by A65, A66, A68, A80, A77, A100, FUNCT_1:def_4;
k2 -' 1 <> 0
proof
assume k2 -' 1 = 0 ; ::_thesis: contradiction
then i1 -' k1 = 0 by A98, A101, RVSUM_1:72;
hence contradiction by A29, A70, A98; ::_thesis: verum
end;
then k2 -' 1 >= 1 by NAT_1:14;
then A102: k2 -' 1 in dom (Len J) by A89, FINSEQ_3:25;
then (Len J) . (k2 -' 1) <> 0 by A7, A78;
hence x1 = x2 by A72, A98, A99, A101, A102, MATRIXJ1:6; ::_thesis: verum
end;
supposeA103: ( i1 -' k1 <> Sum ((Len J) | (k1 -' 1)) & i2 -' k2 <> Sum ((Len J) | (k2 -' 1)) ) ; ::_thesis: x1 = x2
then A104: min ((Len J),(i2 -' k2)) = k2 by A57, A82;
A105: ( F . v2 = b1 . ((i2 -' k2) + 1) & (i2 -' k2) + 1 in dom b1 ) by A57, A82, A83, A84, A103;
( F . v1 = b1 . ((i1 -' k1) + 1) & (i1 -' k1) + 1 in dom b1 ) by A57, A70, A71, A72, A103;
then (i1 -' k1) + 1 = (i2 -' k2) + 1 by A65, A66, A68, A80, A77, A105, FUNCT_1:def_4;
then i1 - k1 = i2 - k1 by A57, A70, A73, A86, A103, A104;
hence x1 = x2 by A68, A71, A80, A83; ::_thesis: verum
end;
end;
end;
A106: RB c= rngb1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in RB or x in rngb1 )
assume x in RB ; ::_thesis: x in rngb1
then consider v1 being Vector of V1 such that
A107: x = v1 and
A108: ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k <> Sum ((Len J) | (k -' 1)) & i -' k = Sum ((Len J) | k) ) ;
consider i, k being Nat such that
A109: ( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k <> Sum ((Len J) | (k -' 1)) ) and
i -' k = Sum ((Len J) | k) by A108;
( v1 = b1 . (i -' k) & i -' k in dom b1 ) by A57, A109;
hence x in rngb1 by A107, FUNCT_1:def_3; ::_thesis: verum
end;
A110: Carrier l c= RB \/ RA
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Carrier l or x in RB \/ RA )
assume A111: x in Carrier l ; ::_thesis: x in RB \/ RA
reconsider v1 = x as Vector of V1 by A111;
Carrier l c= RNG by VECTSP_6:def_4;
then consider i being set such that
A112: i in dom B and
A113: B . i = v1 by A111, FUNCT_1:def_3;
reconsider i = i as Nat by A112;
r . i = r /. i by A30, A59, A112, PARTFUN1:def_6;
then reconsider k = r . i as Element of NAT ;
A114: i -' k <= Sum ((Len J) | k) by A29, A59, A112;
percases ( i -' k = Sum ((Len J) | k) or i -' k < Sum ((Len J) | k) ) by A114, XXREAL_0:1;
supposeA115: i -' k = Sum ((Len J) | k) ; ::_thesis: x in RB \/ RA
A116: S2[i,k] by A29, A59, A112;
then 1 <= k by FINSEQ_3:25;
then k -' 1 = k - 1 by XREAL_1:233;
then (k -' 1) + 1 = k ;
then (Len J) | k = ((Len J) | (k -' 1)) ^ <*((Len J) . k)*> by A116, FINSEQ_5:10;
then ( dom (Len J) = dom J & i -' k = (Sum ((Len J) | (k -' 1))) + ((Len J) . k) ) by A115, MATRIXJ1:def_3, RVSUM_1:74;
then i -' k <> Sum ((Len J) | (k -' 1)) by A7, A116;
then ( v1 in RB or v1 in RA ) by A59, A112, A113, A115;
hence x in RB \/ RA by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose i -' k < Sum ((Len J) | k) ; ::_thesis: x in RB \/ RA
then ( v1 in RB or v1 in RA ) by A59, A112, A113;
hence x in RB \/ RA by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
F .: RA c= rngb1
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in F .: RA or y in rngb1 )
assume y in F .: RA ; ::_thesis: y in rngb1
then consider x being set such that
x in dom F and
A117: x in RA and
A118: y = F . x by FUNCT_1:def_6;
consider v1 being Vector of V1 such that
A119: x = v1 and
A120: ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k < Sum ((Len J) | k) ) by A117;
consider i, k being Nat such that
A121: ( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k < Sum ((Len J) | k) ) by A120;
( i -' k <> Sum ((Len J) | (k -' 1)) or i -' k = Sum ((Len J) | (k -' 1)) ) ;
then ( F . v1 = b1 . ((i -' k) + 1) & (i -' k) + 1 in dom b1 ) by A57, A121;
hence y in rngb1 by A118, A119, FUNCT_1:def_3; ::_thesis: verum
end;
then A122: F .: RA is linearly-independent Subset of V1 by VECTSP_7:1;
F .: RB c= {(0. V1)}
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in F .: RB or y in {(0. V1)} )
assume y in F .: RB ; ::_thesis: y in {(0. V1)}
then consider x being set such that
x in dom F and
A123: x in RB and
A124: y = F . x by FUNCT_1:def_6;
consider v1 being Vector of V1 such that
A125: x = v1 and
A126: ex i, k being Nat st
( i in Seg ((len J) + (Sum (Len J))) & k = r . i & v1 = B . i & i -' k <> Sum ((Len J) | (k -' 1)) & i -' k = Sum ((Len J) | k) ) by A123;
F . v1 = 0. V1 by A57, A126;
hence y in {(0. V1)} by A124, A125, TARSKI:def_1; ::_thesis: verum
end;
then Carrier l c= RB by A61, A110, A62, A122, VECTSP11:44;
then Carrier l c= rngb1 by A106, XBOOLE_1:1;
then l is Linear_Combination of rngb1 by VECTSP_6:def_4;
hence Carrier l = {} by A61, VECTSP_7:def_1; ::_thesis: verum
end;
then A127: RNG is linearly-independent Subset of V1 by VECTSP_7:def_1;
reconsider BAS = the Basis of V1, RNG = RNG as finite Subset of V1 by A4, VECTSP_9:21;
consider C being finite Subset of V1 such that
C c= BAS and
A128: card C = (card BAS) - (card RNG) and
A129: VectSpStr(# the carrier of V1, the addF of V1, the ZeroF of V1, the lmult of V1 #) = Lin (RNG \/ C) by A127, A5, VECTSP_9:19;
A130: (Omega). (Lin BAS) = (Omega). V1 by VECTSP_7:def_3;
then A131: dim V1 = dim (Lin BAS) by VECTSP_9:28;
defpred S5[ Nat] means ( $1 <= card C implies ex f being FinSequence of V1 st
( f is one-to-one & len f = card C & RNG misses rng f & RNG \/ (rng f) is Basis of V1 & ( for i being Nat st i in dom f & i <= $1 holds
F . (f . i) = 0. V1 ) ) );
A132: for n being Nat st S5[n] holds
S5[n + 1]
proof
let n be Nat; ::_thesis: ( S5[n] implies S5[n + 1] )
assume A133: S5[n] ; ::_thesis: S5[n + 1]
set n1 = n + 1;
assume A134: n + 1 <= card C ; ::_thesis: ex f being FinSequence of V1 st
( f is one-to-one & len f = card C & RNG misses rng f & RNG \/ (rng f) is Basis of V1 & ( for i being Nat st i in dom f & i <= n + 1 holds
F . (f . i) = 0. V1 ) )
then consider f being FinSequence of V1 such that
A135: f is one-to-one and
A136: len f = card C and
A137: RNG misses rng f and
A138: RNG \/ (rng f) is Basis of V1 and
A139: for i being Nat st i in dom f & i <= n holds
F . (f . i) = 0. V1 by A133, NAT_1:13;
percases ( F . (f . (n + 1)) = 0. V1 or F . (f . (n + 1)) <> 0. V1 ) ;
suppose F . (f . (n + 1)) = 0. V1 ; ::_thesis: ex f being FinSequence of V1 st
( f is one-to-one & len f = card C & RNG misses rng f & RNG \/ (rng f) is Basis of V1 & ( for i being Nat st i in dom f & i <= n + 1 holds
F . (f . i) = 0. V1 ) )
then for i being Nat st i in dom f & i <= n + 1 holds
F . (f . i) = 0. V1 by A139, NAT_1:8;
hence ex f being FinSequence of V1 st
( f is one-to-one & len f = card C & RNG misses rng f & RNG \/ (rng f) is Basis of V1 & ( for i being Nat st i in dom f & i <= n + 1 holds
F . (f . i) = 0. V1 ) ) by A135, A136, A137, A138; ::_thesis: verum
end;
supposeA140: F . (f . (n + 1)) <> 0. V1 ; ::_thesis: ex f being FinSequence of V1 st
( f is one-to-one & len f = card C & RNG misses rng f & RNG \/ (rng f) is Basis of V1 & ( for i being Nat st i in dom f & i <= n + 1 holds
F . (f . i) = 0. V1 ) )
reconsider Rf = RNG \/ (rng f) as finite Subset of V1 by A138;
reconsider rngB1 = rng b1 as Basis of im (F |^ 1) by MATRLIN:def_2;
set fn = f /. (n + 1);
1 <= n + 1 by NAT_1:14;
then A141: n + 1 in dom f by A134, A136, FINSEQ_3:25;
then A142: f /. (n + 1) = f . (n + 1) by PARTFUN1:def_6;
A143: rng b1 c= F .: RNG
proof
A144: dom F = [#] V1 by FUNCT_2:def_1;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng b1 or y in F .: RNG )
assume y in rng b1 ; ::_thesis: y in F .: RNG
then consider x being set such that
A145: x in dom b1 and
A146: b1 . x = y by FUNCT_1:def_3;
reconsider x = x as Element of NAT by A145;
A147: ( len (Len J) = len J & x <= Sum (Len J) ) by A8, A145, CARD_1:def_7, FINSEQ_3:25;
set m = min ((Len J),x);
A148: x <= Sum ((Len J) | (min ((Len J),x))) by A9, A145, MATRIXJ1:def_1;
A149: min ((Len J),x) in dom (Len J) by A9, A145, MATRIXJ1:def_1;
then min ((Len J),x) <= len (Len J) by FINSEQ_3:25;
then (min ((Len J),x)) + x <= (len J) + (Sum (Len J)) by A147, XREAL_1:7;
then A150: ((min ((Len J),x)) + x) - 1 <= ((len J) + (Sum (Len J))) - 1 by XREAL_1:9;
set x1 = x -' 1;
A151: 1 <= x by A145, FINSEQ_3:25;
then A152: x -' 1 = x - 1 by XREAL_1:233;
1 <= min ((Len J),x) by A149, FINSEQ_3:25;
then 1 + 1 <= (min ((Len J),x)) + x by A151, XREAL_1:7;
then A153: 2 - 1 <= ((min ((Len J),x)) + x) - 1 by XREAL_1:9;
set mx = (min ((Len J),x)) + (x -' 1);
A154: ((min ((Len J),x)) + (x -' 1)) -' (min ((Len J),x)) = ((min ((Len J),x)) + (x -' 1)) - (min ((Len J),x)) by NAT_1:11, XREAL_1:233;
((len J) + (Sum (Len J))) - 1 <= ((len J) + (Sum (Len J))) - 0 by XREAL_1:10;
then (min ((Len J),x)) + (x -' 1) <= (len J) + (Sum (Len J)) by A152, A150, XXREAL_0:2;
then A155: (min ((Len J),x)) + (x -' 1) in Seg ((Sum (Len J)) + (len J)) by A152, A153;
then r . ((min ((Len J),x)) + (x -' 1)) = r /. ((min ((Len J),x)) + (x -' 1)) by A30, PARTFUN1:def_6;
then reconsider k = r . ((min ((Len J),x)) + (x -' 1)) as Element of NAT ;
A156: B . ((min ((Len J),x)) + (x -' 1)) in RNG by A59, A155, FUNCT_1:def_3;
A157: Sum ((Len J) | ((min ((Len J),x)) -' 1)) < (x -' 1) + 1 by A9, A145, A152, MATRIXJ1:7;
then ( min ((Len J),x) <= (min ((Len J),x)) + (x -' 1) & Sum ((Len J) | ((min ((Len J),x)) -' 1)) <= ((min ((Len J),x)) + (x -' 1)) -' (min ((Len J),x)) ) by A154, NAT_1:11, NAT_1:13;
then A158: min ((Len J),x) <= k by A29, A149, A155;
A159: min ((Len J),x) = k
proof
assume min ((Len J),x) <> k ; ::_thesis: contradiction
then A160: min ((Len J),x) < k by A158, XXREAL_0:1;
then reconsider k1 = k - 1 as Element of NAT by NAT_1:20;
A161: k = k1 + 1 ;
then min ((Len J),x) <= k1 by A160, NAT_1:13;
then A162: Sum ((Len J) | (min ((Len J),x))) <= Sum ((Len J) | k1) by POLYNOM3:18;
A163: ((min ((Len J),x)) + (x -' 1)) -' k <= ((min ((Len J),x)) + (x -' 1)) -' (min ((Len J),x)) by A158, NAT_D:41;
k -' 1 = k1 by A161, NAT_D:34;
then Sum ((Len J) | k1) <= ((min ((Len J),x)) + (x -' 1)) -' k by A29, A155;
then Sum ((Len J) | (min ((Len J),x))) <= ((min ((Len J),x)) + (x -' 1)) -' k by A162, XXREAL_0:2;
then Sum ((Len J) | (min ((Len J),x))) <= x -' 1 by A154, A163, XXREAL_0:2;
hence contradiction by A152, A157, A148, NAT_1:13; ::_thesis: verum
end;
A164: ( ((min ((Len J),x)) + (x -' 1)) -' (min ((Len J),x)) = Sum ((Len J) | ((min ((Len J),x)) -' 1)) or ((min ((Len J),x)) + (x -' 1)) -' (min ((Len J),x)) <> Sum ((Len J) | ((min ((Len J),x)) -' 1)) ) ;
((min ((Len J),x)) + (x -' 1)) -' (min ((Len J),x)) < Sum ((Len J) | (min ((Len J),x))) by A152, A154, A157, A148, NAT_1:13;
then F . (B . ((min ((Len J),x)) + (x -' 1))) = b1 . ((((min ((Len J),x)) + (x -' 1)) -' (min ((Len J),x))) + 1) by A57, A155, A159, A164;
hence y in F .: RNG by A146, A152, A154, A156, A144, FUNCT_1:def_6; ::_thesis: verum
end;
( F . (f /. (n + 1)) in im F & F |^ 1 = F ) by RANKNULL:13, VECTSP11:19;
then F . (f /. (n + 1)) in Lin rngB1 by VECTSP_7:def_3;
then consider L being Linear_Combination of rngB1 such that
A165: F . (f /. (n + 1)) = Sum L by VECTSP_7:7;
consider K being Linear_Combination of V1 such that
A166: Carrier L = Carrier K and
A167: Sum L = Sum K by VECTSP_9:8;
Carrier L c= rngB1 by VECTSP_6:def_4;
then consider M being Linear_Combination of RNG such that
A168: F . (Sum M) = Sum K by A143, A166, VECTSP11:41, XBOOLE_1:1;
A169: f . (n + 1) in rng f by A141, FUNCT_1:def_3;
then A170: f /. (n + 1) in Rf by A142, XBOOLE_0:def_3;
A171: not f /. (n + 1) in RNG by A137, A142, A169, XBOOLE_0:3;
not f /. (n + 1) in RNG by A137, A142, A169, XBOOLE_0:3;
then A172: RNG c= Rf \ {(f /. (n + 1))} by XBOOLE_1:7, ZFMISC_1:34;
( Carrier M c= RNG & Carrier M = Carrier (- M) ) by VECTSP_6:38, VECTSP_6:def_4;
then Carrier (- M) c= Rf \ {(f /. (n + 1))} by A172, XBOOLE_1:1;
then reconsider M9 = - M as Linear_Combination of Rf \ {(f /. (n + 1))} by VECTSP_6:def_4;
set fnM = (f /. (n + 1)) + (Sum M9);
A173: (f /. (n + 1)) + (Sum M9) <> f /. (n + 1)
proof
assume (f /. (n + 1)) + (Sum M9) = f /. (n + 1) ; ::_thesis: contradiction
then 0. V1 = ((f /. (n + 1)) + (Sum M9)) - (f /. (n + 1)) by VECTSP_1:16
.= (Sum M9) + ((f /. (n + 1)) - (f /. (n + 1))) by RLVECT_1:def_3
.= (Sum M9) + (0. V1) by RLVECT_1:def_10
.= Sum M9 by RLVECT_1:def_4
.= - (Sum M) by VECTSP_6:46 ;
then 0. V1 = Sum M by VECTSP_1:28;
hence contradiction by A140, A142, A165, A167, A168, RANKNULL:9; ::_thesis: verum
end;
take ff = f +* ((n + 1),((f /. (n + 1)) + (Sum M9))); ::_thesis: ( ff is one-to-one & len ff = card C & RNG misses rng ff & RNG \/ (rng ff) is Basis of V1 & ( for i being Nat st i in dom ff & i <= n + 1 holds
F . (ff . i) = 0. V1 ) )
set fnS = (n + 1) .--> ((f /. (n + 1)) + (Sum M9));
A174: Rf is linearly-independent by A138, VECTSP_7:def_3;
A175: not (f /. (n + 1)) + (Sum M9) in Rf \ {(f /. (n + 1))}
proof
card Rf <> 0 by A169;
then reconsider c1 = (card Rf) - 1 as Element of NAT by NAT_1:20;
assume (f /. (n + 1)) + (Sum M9) in Rf \ {(f /. (n + 1))} ; ::_thesis: contradiction
then A176: (Rf \ {(f /. (n + 1))}) \/ {((f /. (n + 1)) + (Sum M9))} = Rf \ {(f /. (n + 1))} by ZFMISC_1:40;
c1 + 1 = card Rf ;
then A177: card (Rf \ {(f /. (n + 1))}) = c1 by A170, STIRL2_1:55;
card ((Rf \ {(f /. (n + 1))}) \/ {((f /. (n + 1)) + (Sum M9))}) = c1 + 1 by A174, A170, VECTSP11:40;
hence contradiction by A177, A176; ::_thesis: verum
end;
not (f /. (n + 1)) + (Sum M9) in rng f
proof
assume (f /. (n + 1)) + (Sum M9) in rng f ; ::_thesis: contradiction
then (f /. (n + 1)) + (Sum M9) in Rf by XBOOLE_0:def_3;
hence contradiction by A175, A173, ZFMISC_1:56; ::_thesis: verum
end;
then A178: rng f misses {((f /. (n + 1)) + (Sum M9))} by ZFMISC_1:50;
rng ((n + 1) .--> ((f /. (n + 1)) + (Sum M9))) = {((f /. (n + 1)) + (Sum M9))} by FUNCOP_1:8;
then f +* ((n + 1) .--> ((f /. (n + 1)) + (Sum M9))) is one-to-one by A135, A178, FUNCT_4:92;
hence ff is one-to-one by A141, FUNCT_7:def_3; ::_thesis: ( len ff = card C & RNG misses rng ff & RNG \/ (rng ff) is Basis of V1 & ( for i being Nat st i in dom ff & i <= n + 1 holds
F . (ff . i) = 0. V1 ) )
A179: dom ff = dom f by FUNCT_7:30;
hence len ff = card C by A136, FINSEQ_3:29; ::_thesis: ( RNG misses rng ff & RNG \/ (rng ff) is Basis of V1 & ( for i being Nat st i in dom ff & i <= n + 1 holds
F . (ff . i) = 0. V1 ) )
A180: rng ff = ((rng f) \ {(f /. (n + 1))}) \/ {((f /. (n + 1)) + (Sum M9))} by A135, A141, A142, Lm1;
thus RNG misses rng ff ::_thesis: ( RNG \/ (rng ff) is Basis of V1 & ( for i being Nat st i in dom ff & i <= n + 1 holds
F . (ff . i) = 0. V1 ) )
proof
assume RNG meets rng ff ; ::_thesis: contradiction
then consider x being set such that
A181: x in RNG and
A182: x in rng ff by XBOOLE_0:3;
not x in (rng f) \ {(f /. (n + 1))} by A137, A181, XBOOLE_0:3;
then x in {((f /. (n + 1)) + (Sum M9))} by A180, A182, XBOOLE_0:def_3;
then A183: x = (f /. (n + 1)) + (Sum M9) by TARSKI:def_1;
not (f /. (n + 1)) + (Sum M9) in Rf by A175, A173, ZFMISC_1:56;
hence contradiction by A181, A183, XBOOLE_0:def_3; ::_thesis: verum
end;
A184: (Rf \ {(f /. (n + 1))}) \/ {((f /. (n + 1)) + (Sum M9))} = ((RNG \ {(f /. (n + 1))}) \/ ((rng f) \ {(f /. (n + 1))})) \/ {((f /. (n + 1)) + (Sum M9))} by XBOOLE_1:42
.= (RNG \/ ((rng f) \ {(f /. (n + 1))})) \/ {((f /. (n + 1)) + (Sum M9))} by A171, ZFMISC_1:57
.= RNG \/ (rng ff) by A180, XBOOLE_1:4 ;
then reconsider Rff = RNG \/ (rng ff) as finite Subset of V1 ;
dim V1 = card Rf by A138, VECTSP_9:def_1
.= card (RNG \/ (rng ff)) by A174, A170, A184, VECTSP11:40 ;
then dim (Lin Rff) = dim V1 by A174, A170, A184, VECTSP11:40, VECTSP_9:26;
then A185: (Omega). V1 = (Omega). (Lin Rff) by VECTSP_9:28;
(Rf \ {(f /. (n + 1))}) \/ {((f /. (n + 1)) + (Sum M9))} is linearly-independent by A174, A170, VECTSP11:40;
hence RNG \/ (rng ff) is Basis of V1 by A184, A185, VECTSP_7:def_3; ::_thesis: for i being Nat st i in dom ff & i <= n + 1 holds
F . (ff . i) = 0. V1
let i be Nat; ::_thesis: ( i in dom ff & i <= n + 1 implies F . (ff . i) = 0. V1 )
assume that
A186: i in dom ff and
A187: i <= n + 1 ; ::_thesis: F . (ff . i) = 0. V1
percases ( i < n + 1 or i = n + 1 ) by A187, XXREAL_0:1;
suppose i < n + 1 ; ::_thesis: F . (ff . i) = 0. V1
then ( ff . i = f . i & i <= n ) by FUNCT_7:32, NAT_1:13;
hence F . (ff . i) = 0. V1 by A139, A179, A186; ::_thesis: verum
end;
suppose i = n + 1 ; ::_thesis: F . (ff . i) = 0. V1
then ff . i = (f /. (n + 1)) + (Sum M9) by A179, A186, FUNCT_7:31;
hence F . (ff . i) = F . ((f /. (n + 1)) - (Sum M)) by VECTSP_6:46
.= (F . (f /. (n + 1))) - (F . (Sum M)) by RANKNULL:8
.= 0. V1 by A165, A167, A168, RLVECT_1:def_10 ;
::_thesis: verum
end;
end;
end;
end;
end;
A188: card (RNG \/ C) = ((card RNG) + (card C)) - (card (RNG /\ C)) by CARD_2:45
.= (card BAS) - (card (RNG /\ C)) by A128 ;
then (card (RNG \/ C)) + (card (RNG /\ C)) = card BAS ;
then A189: card (RNG \/ C) <= card BAS by NAT_1:11;
A190: dim (Lin BAS) = card BAS by A4, VECTSP_9:26;
then A191: card (RNG \/ C) >= card BAS by A130, A129, MATRLIN2:6;
then A192: card (RNG \/ C) = card BAS by A189, XXREAL_0:1;
dim V1 = dim (Lin (RNG \/ C)) by A130, A129, VECTSP_9:28;
then A193: RNG \/ C is linearly-independent by A131, A190, A191, A189, MATRLIN2:5, XXREAL_0:1;
A194: S5[ 0 ]
proof
assume 0 <= card C ; ::_thesis: ex f being FinSequence of V1 st
( f is one-to-one & len f = card C & RNG misses rng f & RNG \/ (rng f) is Basis of V1 & ( for i being Nat st i in dom f & i <= 0 holds
F . (f . i) = 0. V1 ) )
card C = card (Seg (card C)) by FINSEQ_1:57;
then Seg (card C),C are_equipotent by CARD_1:5;
then consider f being Function such that
A195: f is one-to-one and
A196: dom f = Seg (card C) and
A197: rng f = C by WELLORD2:def_4;
reconsider f = f as FinSequence by A196, FINSEQ_1:def_2;
reconsider f = f as FinSequence of V1 by A197, FINSEQ_1:def_4;
take f ; ::_thesis: ( f is one-to-one & len f = card C & RNG misses rng f & RNG \/ (rng f) is Basis of V1 & ( for i being Nat st i in dom f & i <= 0 holds
F . (f . i) = 0. V1 ) )
thus ( f is one-to-one & len f = card C ) by A195, A196, FINSEQ_1:def_3; ::_thesis: ( RNG misses rng f & RNG \/ (rng f) is Basis of V1 & ( for i being Nat st i in dom f & i <= 0 holds
F . (f . i) = 0. V1 ) )
RNG /\ C = {} by A188, A192;
hence ( RNG misses rng f & RNG \/ (rng f) is Basis of V1 ) by A129, A193, A197, VECTSP_7:def_3, XBOOLE_0:def_7; ::_thesis: for i being Nat st i in dom f & i <= 0 holds
F . (f . i) = 0. V1
let i be Nat; ::_thesis: ( i in dom f & i <= 0 implies F . (f . i) = 0. V1 )
assume ( i in dom f & i <= 0 ) ; ::_thesis: F . (f . i) = 0. V1
hence F . (f . i) = 0. V1 by FINSEQ_3:25; ::_thesis: verum
end;
for n being Nat holds S5[n] from NAT_1:sch_2(A194, A132);
then consider f being FinSequence of V1 such that
A198: f is one-to-one and
A199: len f = card C and
A200: RNG misses rng f and
A201: RNG \/ (rng f) is Basis of V1 and
A202: for i being Nat st i in dom f & i <= card C holds
F . (f . i) = 0. V1 ;
A203: rng (B ^ f) = (rng B) \/ (rng f) by FINSEQ_1:31;
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_dom_B_&_x2_in_dom_B_&_B_._x1_=_B_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in dom B & x2 in dom B & B . x1 = B . x2 implies x1 = x2 )
assume that
A204: x1 in dom B and
A205: x2 in dom B and
A206: B . x1 = B . x2 ; ::_thesis: x1 = x2
reconsider i1 = x1, i2 = x2 as Nat by A204, A205;
( r /. i1 = r . i1 & r /. i2 = r . i2 ) by A30, A59, A204, A205, PARTFUN1:def_6;
then reconsider k1 = r . i1, k2 = r . i2 as Element of NAT ;
A207: ( i1 -' k1 = Sum ((Len J) | (k1 -' 1)) implies ( F . (B . x1) = b1 . ((i1 -' k1) + 1) & (i1 -' k1) + 1 in dom b1 ) ) by A57, A59, A204;
A208: S2[i1,k1] by A29, A59, A204;
then A209: i1 -' k1 = i1 - k1 by XREAL_1:233;
A210: S2[i2,k2] by A29, A59, A205;
then A211: i2 -' k2 = i2 - k2 by XREAL_1:233;
A212: k2 -' 1 <= k2 by NAT_D:35;
A213: ( i1 -' k1 <> Sum ((Len J) | (k1 -' 1)) implies ( B . x1 = b1 . (i1 -' k1) & i1 -' k1 in dom b1 & min ((Len J),(i1 -' k1)) = k1 & ( i1 -' k1 < Sum ((Len J) | k1) implies ( F . (B . x1) = b1 . ((i1 -' k1) + 1) & (i1 -' k1) + 1 in dom b1 ) ) & ( i1 -' k1 = Sum ((Len J) | k1) implies F . (B . x1) = 0. V1 ) ) ) by A57, A59, A204;
k2 <= len (Len J) by A210, FINSEQ_3:25;
then A214: k2 -' 1 <= len (Len J) by A212, XXREAL_0:2;
1 <= k1 by A208, FINSEQ_3:25;
then A215: k1 -' 1 = k1 - 1 by XREAL_1:233;
A216: ( i2 -' k2 <> Sum ((Len J) | (k2 -' 1)) implies ( B . x2 = b1 . (i2 -' k2) & i2 -' k2 in dom b1 & min ((Len J),(i2 -' k2)) = k2 & ( i2 -' k2 < Sum ((Len J) | k2) implies ( F . (B . x2) = b1 . ((i2 -' k2) + 1) & (i2 -' k2) + 1 in dom b1 ) ) & ( i2 -' k2 = Sum ((Len J) | k2) implies F . (B . x2) = 0. V1 ) ) ) by A57, A59, A205;
1 <= k2 by A210, FINSEQ_3:25;
then A217: k2 -' 1 = k2 - 1 by XREAL_1:233;
A218: ( i2 -' k2 = Sum ((Len J) | (k2 -' 1)) implies ( F . (B . x2) = b1 . ((i2 -' k2) + 1) & (i2 -' k2) + 1 in dom b1 ) ) by A57, A59, A205;
A219: k1 -' 1 <= k1 by NAT_D:35;
k1 <= len (Len J) by A208, FINSEQ_3:25;
then A220: k1 -' 1 <= len (Len J) by A219, XXREAL_0:2;
A221: dom (Len J) = dom J by MATRIXJ1:def_3;
rng b1 is Basis of im (F |^ 1) by MATRLIN:def_2;
then A222: rng b1 is linearly-independent Subset of (im (F |^ 1)) by VECTSP_7:def_3;
A223: b1 is one-to-one by MATRLIN:def_2;
A224: ( i1 -' k1 <= Sum ((Len J) | k1) & i2 -' k2 <= Sum ((Len J) | k2) ) by A29, A59, A204, A205;
now__::_thesis:_i1_=_i2
percases ( ( i1 -' k1 = Sum ((Len J) | (k1 -' 1)) & i2 -' k2 = Sum ((Len J) | (k2 -' 1)) ) or ( i1 -' k1 = Sum ((Len J) | (k1 -' 1)) & i2 -' k2 <> Sum ((Len J) | (k2 -' 1)) & i2 -' k2 < Sum ((Len J) | k2) ) or ( i1 -' k1 = Sum ((Len J) | (k1 -' 1)) & i2 -' k2 <> Sum ((Len J) | (k2 -' 1)) & i2 -' k2 = Sum ((Len J) | k2) ) or ( i2 -' k2 = Sum ((Len J) | (k2 -' 1)) & i1 -' k1 <> Sum ((Len J) | (k1 -' 1)) & i1 -' k1 < Sum ((Len J) | k1) ) or ( i2 -' k2 = Sum ((Len J) | (k2 -' 1)) & i1 -' k1 <> Sum ((Len J) | (k1 -' 1)) & i1 -' k1 = Sum ((Len J) | k1) ) or ( i1 -' k1 <> Sum ((Len J) | (k1 -' 1)) & i2 -' k2 <> Sum ((Len J) | (k2 -' 1)) ) ) by A224, XXREAL_0:1;
supposeA225: ( i1 -' k1 = Sum ((Len J) | (k1 -' 1)) & i2 -' k2 = Sum ((Len J) | (k2 -' 1)) ) ; ::_thesis: i1 = i2
then A226: ( F . (B . x2) = b1 . ((i2 -' k2) + 1) & (i2 -' k2) + 1 in dom b1 ) by A57, A59, A205;
( F . (B . x1) = b1 . ((i1 -' k1) + 1) & (i1 -' k1) + 1 in dom b1 ) by A57, A59, A204, A225;
then A227: (i1 -' k1) + 1 = (i2 -' k2) + 1 by A206, A223, A226, FUNCT_1:def_4;
A228: ( k2 -' 1 in dom (Len J) implies (Len J) . (k2 -' 1) <> 0 ) by A7, A221;
( k1 -' 1 in dom (Len J) implies (Len J) . (k1 -' 1) <> 0 ) by A7, A221;
then k1 -' 1 = k2 -' 1 by A220, A214, A225, A227, A228, MATRIXJ1:11;
hence i1 = i2 by A215, A217, A209, A211, A227; ::_thesis: verum
end;
supposeA229: ( i1 -' k1 = Sum ((Len J) | (k1 -' 1)) & i2 -' k2 <> Sum ((Len J) | (k2 -' 1)) & i2 -' k2 < Sum ((Len J) | k2) ) ; ::_thesis: i1 = i2
then A230: min ((Len J),(i2 -' k2)) = k2 by A57, A59, A205;
A231: ( F . (B . x2) = b1 . ((i2 -' k2) + 1) & (i2 -' k2) + 1 in dom b1 ) by A57, A59, A205, A229;
( F . (B . x1) = b1 . ((i1 -' k1) + 1) & (i1 -' k1) + 1 in dom b1 ) by A57, A59, A204, A229;
then A232: (i1 -' k1) + 1 = (i2 -' k2) + 1 by A206, A223, A231, FUNCT_1:def_4;
k1 -' 1 <> 0
proof
assume k1 -' 1 = 0 ; ::_thesis: contradiction
then (Len J) | (k1 -' 1) = <*> REAL ;
hence contradiction by A29, A59, A205, A229, A232, RVSUM_1:72; ::_thesis: verum
end;
then k1 -' 1 >= 1 by NAT_1:14;
then A233: k1 -' 1 in dom (Len J) by A220, FINSEQ_3:25;
then (Len J) . (k1 -' 1) <> 0 by A7, A221;
hence i1 = i2 by A229, A230, A232, A233, MATRIXJ1:6; ::_thesis: verum
end;
suppose ( i1 -' k1 = Sum ((Len J) | (k1 -' 1)) & i2 -' k2 <> Sum ((Len J) | (k2 -' 1)) & i2 -' k2 = Sum ((Len J) | k2) ) ; ::_thesis: i1 = i2
then ( b1 . ((i1 -' k1) + 1) = 0. (im (F |^ 1)) & b1 . ((i1 -' k1) + 1) in rng b1 ) by A206, A207, A216, FUNCT_1:def_3, VECTSP_4:11;
hence i1 = i2 by A222, VECTSP_7:2; ::_thesis: verum
end;
supposeA234: ( i2 -' k2 = Sum ((Len J) | (k2 -' 1)) & i1 -' k1 <> Sum ((Len J) | (k1 -' 1)) & i1 -' k1 < Sum ((Len J) | k1) ) ; ::_thesis: i1 = i2
then A235: min ((Len J),(i1 -' k1)) = k1 by A57, A59, A204;
A236: ( F . (B . x1) = b1 . ((i1 -' k1) + 1) & (i1 -' k1) + 1 in dom b1 ) by A57, A59, A204, A234;
( F . (B . x2) = b1 . ((i2 -' k2) + 1) & (i2 -' k2) + 1 in dom b1 ) by A57, A59, A205, A234;
then A237: (i2 -' k2) + 1 = (i1 -' k1) + 1 by A206, A223, A236, FUNCT_1:def_4;
k2 -' 1 <> 0
proof
assume k2 -' 1 = 0 ; ::_thesis: contradiction
then i1 -' k1 = 0 by A234, A237, RVSUM_1:72;
hence contradiction by A29, A59, A204, A234; ::_thesis: verum
end;
then k2 -' 1 >= 1 by NAT_1:14;
then A238: k2 -' 1 in dom (Len J) by A214, FINSEQ_3:25;
then (Len J) . (k2 -' 1) <> 0 by A7, A221;
hence i1 = i2 by A234, A235, A237, A238, MATRIXJ1:6; ::_thesis: verum
end;
suppose ( i2 -' k2 = Sum ((Len J) | (k2 -' 1)) & i1 -' k1 <> Sum ((Len J) | (k1 -' 1)) & i1 -' k1 = Sum ((Len J) | k1) ) ; ::_thesis: i1 = i2
then ( b1 . ((i2 -' k2) + 1) = 0. (im (F |^ 1)) & b1 . ((i2 -' k2) + 1) in rng b1 ) by A206, A213, A218, FUNCT_1:def_3, VECTSP_4:11;
hence i1 = i2 by A222, VECTSP_7:2; ::_thesis: verum
end;
supposeA239: ( i1 -' k1 <> Sum ((Len J) | (k1 -' 1)) & i2 -' k2 <> Sum ((Len J) | (k2 -' 1)) ) ; ::_thesis: i1 = i2
then i2 -' k2 = i1 -' k1 by A206, A213, A216, A223, FUNCT_1:def_4;
then i2 - k1 = i1 - k1 by A57, A59, A205, A213, A209, A211, A239;
hence i1 = i2 ; ::_thesis: verum
end;
end;
end;
hence x1 = x2 ; ::_thesis: verum
end;
then B is one-to-one by FUNCT_1:def_4;
then B ^ f is one-to-one by A198, A200, FINSEQ_3:91;
then reconsider Bf = B ^ f as OrdBasis of V1 by A201, A203, MATRLIN:def_2;
for i being Nat holds
( not i in dom Bf or F . (Bf /. i) = (0. K) * (Bf /. i) or ( i + 1 in dom Bf & F . (Bf /. i) = ((0. K) * (Bf /. i)) + (Bf /. (i + 1)) ) )
proof
let i be Nat; ::_thesis: ( not i in dom Bf or F . (Bf /. i) = (0. K) * (Bf /. i) or ( i + 1 in dom Bf & F . (Bf /. i) = ((0. K) * (Bf /. i)) + (Bf /. (i + 1)) ) )
assume A240: i in dom Bf ; ::_thesis: ( F . (Bf /. i) = (0. K) * (Bf /. i) or ( i + 1 in dom Bf & F . (Bf /. i) = ((0. K) * (Bf /. i)) + (Bf /. (i + 1)) ) )
A241: Bf /. i = Bf . i by A240, PARTFUN1:def_6;
percases ( i in dom B or ex j being Nat st
( j in dom f & i = (len B) + j ) ) by A240, FINSEQ_1:25;
supposeA242: i in dom B ; ::_thesis: ( F . (Bf /. i) = (0. K) * (Bf /. i) or ( i + 1 in dom Bf & F . (Bf /. i) = ((0. K) * (Bf /. i)) + (Bf /. (i + 1)) ) )
then r /. i = r . i by A30, A59, PARTFUN1:def_6;
then reconsider k = r . i as Element of NAT ;
A243: i -' k <= Sum ((Len J) | k) by A29, A59, A242;
A244: S2[i,k] by A29, A59, A242;
then 1 <= k by FINSEQ_3:25;
then k -' 1 = k - 1 by XREAL_1:233;
then (k -' 1) + 1 = k ;
then (Len J) | k = ((Len J) | (k -' 1)) ^ <*((Len J) . k)*> by A244, FINSEQ_5:10;
then A245: ( dom (Len J) = dom J & Sum ((Len J) | k) = (Sum ((Len J) | (k -' 1))) + ((Len J) . k) ) by MATRIXJ1:def_3, RVSUM_1:74;
percases ( i -' k = Sum ((Len J) | k) or i -' k < Sum ((Len J) | k) ) by A243, XXREAL_0:1;
supposeA246: i -' k = Sum ((Len J) | k) ; ::_thesis: ( F . (Bf /. i) = (0. K) * (Bf /. i) or ( i + 1 in dom Bf & F . (Bf /. i) = ((0. K) * (Bf /. i)) + (Bf /. (i + 1)) ) )
then A247: i -' k <> Sum ((Len J) | (k -' 1)) by A7, A244, A245;
F . (Bf /. i) = F . (B . i) by A241, A242, FINSEQ_1:def_7
.= 0. V1 by A57, A59, A242, A246, A247
.= (0. K) * (Bf /. i) by VECTSP_1:14 ;
hence ( F . (Bf /. i) = (0. K) * (Bf /. i) or ( i + 1 in dom Bf & F . (Bf /. i) = ((0. K) * (Bf /. i)) + (Bf /. (i + 1)) ) ) ; ::_thesis: verum
end;
supposeA248: i -' k < Sum ((Len J) | k) ; ::_thesis: ( F . (Bf /. i) = (0. K) * (Bf /. i) or ( i + 1 in dom Bf & F . (Bf /. i) = ((0. K) * (Bf /. i)) + (Bf /. (i + 1)) ) )
A249: ( i -' k = Sum ((Len J) | (k -' 1)) or i -' k <> Sum ((Len J) | (k -' 1)) ) ;
then A250: F . (B . i) = b1 . ((i -' k) + 1) by A57, A59, A242, A248;
dom J = dom (Len J) by MATRIXJ1:def_3;
then A251: k <= len J by A244, FINSEQ_3:25;
A252: (i -' k) + 1 <= Sum ((Len J) | k) by A248, NAT_1:13;
A253: i -' k = i - k by A244, XREAL_1:233;
A254: (i -' k) + 1 in dom b1 by A57, A59, A242, A248, A249;
then A255: 1 <= (i -' k) + 1 by FINSEQ_3:25;
then A256: 1 + 0 <= ((i - k) + 1) + k by A253, XREAL_1:7;
(i -' k) + 1 <= Sum (Len J) by A8, A254, FINSEQ_3:25;
then ((i - k) + 1) + k <= (Sum (Len J)) + (len J) by A251, A253, XREAL_1:7;
then A257: i + 1 in Seg ((Sum (Len J)) + (len J)) by A256;
then r /. (i + 1) = r . (i + 1) by A30, PARTFUN1:def_6;
then reconsider k1 = r . (i + 1) as Element of NAT ;
set i1 = i + 1;
A258: dom B c= dom Bf by FINSEQ_1:26;
1 + k <= ((i - k) + 1) + k by A255, A253, XREAL_1:7;
then A259: k <= i + 1 by NAT_1:13;
then A260: (i + 1) -' k = (i + 1) - k by XREAL_1:233;
Sum ((Len J) | (k -' 1)) <= (i -' k) + 1 by A244, NAT_1:12;
then A261: k <= k1 by A29, A244, A253, A257, A259, A260;
A262: S2[i + 1,k1] by A29, A257;
A263: k = k1
proof
assume A264: k <> k1 ; ::_thesis: contradiction
then A265: k < k1 by A261, XXREAL_0:1;
then reconsider K1 = k1 - 1 as Element of NAT by NAT_1:20;
A266: (i + 1) -' k1 <= (i + 1) -' k by A261, NAT_D:41;
(i + 1) - k1 = (i + 1) -' k1 by A262, XREAL_1:233;
then (i + 1) -' k1 <> (i + 1) -' k by A260, A264;
then A267: (i + 1) -' k1 < (i + 1) -' k by A266, XXREAL_0:1;
A268: k1 = K1 + 1 ;
then k <= K1 by A265, NAT_1:13;
then A269: Sum ((Len J) | k) <= Sum ((Len J) | K1) by POLYNOM3:18;
k1 -' 1 = K1 by A268, NAT_D:34;
then Sum ((Len J) | K1) <= (i + 1) -' k1 by A29, A257;
then Sum ((Len J) | k) <= (i + 1) -' k1 by A269, XXREAL_0:2;
hence contradiction by A252, A253, A260, A267, XXREAL_0:2; ::_thesis: verum
end;
Sum ((Len J) | (k -' 1)) < (i -' k) + 1 by A244, NAT_1:13;
then B . (i + 1) = b1 . ((i -' k) + 1) by A57, A253, A257, A260, A263;
then Bf . (i + 1) = b1 . ((i -' k) + 1) by A59, A257, FINSEQ_1:def_7;
then Bf /. (i + 1) = b1 . ((i -' k) + 1) by A59, A257, A258, PARTFUN1:def_6;
then F . (Bf /. i) = Bf /. (i + 1) by A241, A242, A250, FINSEQ_1:def_7
.= (0. V1) + (Bf /. (i + 1)) by RLVECT_1:def_4
.= ((0. K) * (Bf /. i)) + (Bf /. (i + 1)) by VECTSP_1:14 ;
hence ( F . (Bf /. i) = (0. K) * (Bf /. i) or ( i + 1 in dom Bf & F . (Bf /. i) = ((0. K) * (Bf /. i)) + (Bf /. (i + 1)) ) ) by A59, A257, A258; ::_thesis: verum
end;
end;
end;
suppose ex j being Nat st
( j in dom f & i = (len B) + j ) ; ::_thesis: ( F . (Bf /. i) = (0. K) * (Bf /. i) or ( i + 1 in dom Bf & F . (Bf /. i) = ((0. K) * (Bf /. i)) + (Bf /. (i + 1)) ) )
then consider j being Nat such that
A270: j in dom f and
A271: i = (len B) + j ;
A272: j <= len f by A270, FINSEQ_3:25;
F . (Bf /. i) = F . (f . j) by A241, A270, A271, FINSEQ_1:def_7
.= 0. V1 by A199, A202, A270, A272
.= (0. K) * (Bf /. i) by VECTSP_1:14 ;
hence ( F . (Bf /. i) = (0. K) * (Bf /. i) or ( i + 1 in dom Bf & F . (Bf /. i) = ((0. K) * (Bf /. i)) + (Bf /. (i + 1)) ) ) ; ::_thesis: verum
end;
end;
end;
then consider J being non-empty FinSequence_of_Jordan_block of 0. K,K such that
A273: AutMt (F,Bf,Bf) = block_diagonal (J,(0. K)) by Th28;
now__::_thesis:_for_i_being_Nat_st_i_in_dom_J_holds_
(Len_J)_._i_<>_0
A274: dom (Len J) = dom J by MATRIXJ1:def_3;
let i be Nat; ::_thesis: ( i in dom J implies (Len J) . i <> 0 )
assume A275: i in dom J ; ::_thesis: (Len J) . i <> 0
J . i <> {} by A275, FUNCT_1:def_9;
hence (Len J) . i <> 0 by A275, A274, MATRIXJ1:def_3; ::_thesis: verum
end;
hence ex J being FinSequence_of_Jordan_block of 0. K,K ex b1 being OrdBasis of V1 st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for i being Nat st i in dom J holds
(Len J) . i <> 0 ) ) by A273; ::_thesis: verum
end;
A276: S1[ 0 ]
proof
reconsider J = {} as FinSequence_of_Jordan_block of 0. K,K by Th10;
let V1 be finite-dimensional VectSp of K; ::_thesis: for F being nilpotent linear-transformation of V1,V1 st deg F = 0 holds
ex J being FinSequence_of_Jordan_block of 0. K,K ex b1 being OrdBasis of V1 st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for i being Nat st i in dom J holds
(Len J) . i <> 0 ) )
set b1 = the OrdBasis of V1;
let F be nilpotent linear-transformation of V1,V1; ::_thesis: ( deg F = 0 implies ex J being FinSequence_of_Jordan_block of 0. K,K ex b1 being OrdBasis of V1 st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for i being Nat st i in dom J holds
(Len J) . i <> 0 ) ) )
assume deg F = 0 ; ::_thesis: ex J being FinSequence_of_Jordan_block of 0. K,K ex b1 being OrdBasis of V1 st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for i being Nat st i in dom J holds
(Len J) . i <> 0 ) )
then [#] V1 = {(0. V1)} by Th15
.= the carrier of ((0). V1) by VECTSP_4:def_3 ;
then (0). V1 = (Omega). V1 by VECTSP_4:29;
then A277: 0 = dim V1 by VECTSP_9:29
.= len the OrdBasis of V1 by MATRLIN2:21
.= len (AutMt (F, the OrdBasis of V1, the OrdBasis of V1)) by MATRIX_1:def_2 ;
take J ; ::_thesis: ex b1 being OrdBasis of V1 st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for i being Nat st i in dom J holds
(Len J) . i <> 0 ) )
take the OrdBasis of V1 ; ::_thesis: ( AutMt (F, the OrdBasis of V1, the OrdBasis of V1) = block_diagonal (J,(0. K)) & ( for i being Nat st i in dom J holds
(Len J) . i <> 0 ) )
thus AutMt (F, the OrdBasis of V1, the OrdBasis of V1) = {} by A277
.= block_diagonal (J,(0. K)) by MATRIXJ1:22 ; ::_thesis: for i being Nat st i in dom J holds
(Len J) . i <> 0
thus for i being Nat st i in dom J holds
(Len J) . i <> 0 ; ::_thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch_2(A276, A1);
then S1[ deg F] ;
then consider J being FinSequence_of_Jordan_block of 0. K,K, b1 being OrdBasis of V1 such that
A278: AutMt (F,b1,b1) = block_diagonal (J,(0. K)) and
A279: for i being Nat st i in dom J holds
(Len J) . i <> 0 ;
now__::_thesis:_for_x_being_set_st_x_in_dom_J_holds_
not_J_._x_is_empty
let x be set ; ::_thesis: ( x in dom J implies not J . x is empty )
assume A280: x in dom J ; ::_thesis: not J . x is empty
reconsider i = x as Element of NAT by A280;
( dom J = dom (Len J) & (Len J) . i <> 0 ) by A279, A280, MATRIXJ1:def_3;
hence not J . x is empty by A280, MATRIXJ1:def_3; ::_thesis: verum
end;
then J is non-empty by FUNCT_1:def_9;
hence ex J being non-empty FinSequence_of_Jordan_block of 0. K,K ex b1 being OrdBasis of V1 st AutMt (F,b1,b1) = block_diagonal (J,(0. K)) by A278; ::_thesis: verum
end;
theorem Th30: :: MATRIXJ2:30
for n being Nat
for K being Field
for L being Element of K
for V being VectSp of K
for F being linear-transformation of V,V
for V1 being finite-dimensional Subspace of V
for F1 being linear-transformation of V1,V1 st V1 = ker ((F + ((- L) * (id V))) |^ n) & F | V1 = F1 holds
ex J being non-empty FinSequence_of_Jordan_block of L,K ex b1 being OrdBasis of V1 st AutMt (F1,b1,b1) = block_diagonal (J,(0. K))
proof
let n be Nat; ::_thesis: for K being Field
for L being Element of K
for V being VectSp of K
for F being linear-transformation of V,V
for V1 being finite-dimensional Subspace of V
for F1 being linear-transformation of V1,V1 st V1 = ker ((F + ((- L) * (id V))) |^ n) & F | V1 = F1 holds
ex J being non-empty FinSequence_of_Jordan_block of L,K ex b1 being OrdBasis of V1 st AutMt (F1,b1,b1) = block_diagonal (J,(0. K))
let K be Field; ::_thesis: for L being Element of K
for V being VectSp of K
for F being linear-transformation of V,V
for V1 being finite-dimensional Subspace of V
for F1 being linear-transformation of V1,V1 st V1 = ker ((F + ((- L) * (id V))) |^ n) & F | V1 = F1 holds
ex J being non-empty FinSequence_of_Jordan_block of L,K ex b1 being OrdBasis of V1 st AutMt (F1,b1,b1) = block_diagonal (J,(0. K))
let L be Element of K; ::_thesis: for V being VectSp of K
for F being linear-transformation of V,V
for V1 being finite-dimensional Subspace of V
for F1 being linear-transformation of V1,V1 st V1 = ker ((F + ((- L) * (id V))) |^ n) & F | V1 = F1 holds
ex J being non-empty FinSequence_of_Jordan_block of L,K ex b1 being OrdBasis of V1 st AutMt (F1,b1,b1) = block_diagonal (J,(0. K))
let V be VectSp of K; ::_thesis: for F being linear-transformation of V,V
for V1 being finite-dimensional Subspace of V
for F1 being linear-transformation of V1,V1 st V1 = ker ((F + ((- L) * (id V))) |^ n) & F | V1 = F1 holds
ex J being non-empty FinSequence_of_Jordan_block of L,K ex b1 being OrdBasis of V1 st AutMt (F1,b1,b1) = block_diagonal (J,(0. K))
let F be linear-transformation of V,V; ::_thesis: for V1 being finite-dimensional Subspace of V
for F1 being linear-transformation of V1,V1 st V1 = ker ((F + ((- L) * (id V))) |^ n) & F | V1 = F1 holds
ex J being non-empty FinSequence_of_Jordan_block of L,K ex b1 being OrdBasis of V1 st AutMt (F1,b1,b1) = block_diagonal (J,(0. K))
let V1 be finite-dimensional Subspace of V; ::_thesis: for F1 being linear-transformation of V1,V1 st V1 = ker ((F + ((- L) * (id V))) |^ n) & F | V1 = F1 holds
ex J being non-empty FinSequence_of_Jordan_block of L,K ex b1 being OrdBasis of V1 st AutMt (F1,b1,b1) = block_diagonal (J,(0. K))
let F1 be linear-transformation of V1,V1; ::_thesis: ( V1 = ker ((F + ((- L) * (id V))) |^ n) & F | V1 = F1 implies ex J being non-empty FinSequence_of_Jordan_block of L,K ex b1 being OrdBasis of V1 st AutMt (F1,b1,b1) = block_diagonal (J,(0. K)) )
assume that
A1: V1 = ker ((F + ((- L) * (id V))) |^ n) and
A2: F | V1 = F1 ; ::_thesis: ex J being non-empty FinSequence_of_Jordan_block of L,K ex b1 being OrdBasis of V1 st AutMt (F1,b1,b1) = block_diagonal (J,(0. K))
set FL = F + ((- L) * (id V));
reconsider FLV = (F + ((- L) * (id V))) | V1 as nilpotent linear-transformation of V1,V1 by A1, Th14;
A3: now__::_thesis:_for_x_being_set_st_x_in_dom_FLV_holds_
FLV_._x_=_(F1_+_((-_L)_*_(id_V1)))_._x
let x be set ; ::_thesis: ( x in dom FLV implies FLV . x = (F1 + ((- L) * (id V1))) . x )
assume x in dom FLV ; ::_thesis: FLV . x = (F1 + ((- L) * (id V1))) . x
then reconsider v1 = x as Vector of V1 by FUNCT_2:def_1;
reconsider v = v1 as Vector of V by VECTSP_4:10;
(id V) . v = v by FUNCT_1:18;
then A4: (- L) * ((id V) . v) = (- L) * ((id V1) . v1) by FUNCT_1:18, VECTSP_4:14;
A5: F . v1 = F1 . v1 by A2, FUNCT_1:49;
thus FLV . x = (F + ((- L) * (id V))) . v by FUNCT_1:49
.= (F . v) + (((- L) * (id V)) . v) by MATRLIN:def_3
.= (F . v) + ((- L) * ((id V) . v)) by MATRLIN:def_4
.= (F1 . v1) + ((- L) * ((id V1) . v1)) by A4, A5, VECTSP_4:13
.= (F1 . v1) + (((- L) * (id V1)) . v1) by MATRLIN:def_4
.= (F1 + ((- L) * (id V1))) . x by MATRLIN:def_3 ; ::_thesis: verum
end;
dom FLV = the carrier of V1 by FUNCT_2:def_1
.= dom (F1 + ((- L) * (id V1))) by FUNCT_2:def_1 ;
then A6: FLV = F1 + ((- L) * (id V1)) by A3, FUNCT_1:2;
consider J being non-empty FinSequence_of_Jordan_block of 0. K,K, b1 being OrdBasis of V1 such that
A7: AutMt (FLV,b1,b1) = block_diagonal (J,(0. K)) by Th29;
L + (0. K) = L by RLVECT_1:def_4;
then reconsider JM = J (+) (mlt (((len J) |-> L),(1. (K,(Len J))))) as FinSequence_of_Jordan_block of L,K by Th12;
now__::_thesis:_for_x_being_set_st_x_in_dom_JM_holds_
not_JM_._x_is_empty
let x be set ; ::_thesis: ( x in dom JM implies not JM . x is empty )
assume A8: x in dom JM ; ::_thesis: not JM . x is empty
reconsider i = x as Nat by A8;
A9: JM . i = (J . i) + ((mlt (((len J) |-> L),(1. (K,(Len J))))) . i) by A8, MATRIXJ1:def_10;
dom JM = dom J by MATRIXJ1:def_10;
then J . i <> {} by A8, FUNCT_1:def_9;
hence not JM . x is empty by A9, MATRIX_3:def_3; ::_thesis: verum
end;
then reconsider JM = JM as non-empty FinSequence_of_Jordan_block of L,K by FUNCT_1:def_9;
take JM ; ::_thesis: ex b1 being OrdBasis of V1 st AutMt (F1,b1,b1) = block_diagonal (JM,(0. K))
take b1 ; ::_thesis: AutMt (F1,b1,b1) = block_diagonal (JM,(0. K))
set Aid = AutMt ((id V1),b1,b1);
set AF1 = AutMt (F1,b1,b1);
A10: now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(AutMt_(F1,b1,b1))_holds_
(((AutMt_(F1,b1,b1))_+_((-_L)_*_(AutMt_((id_V1),b1,b1))))_+_(L_*_(AutMt_((id_V1),b1,b1))))_*_(i,j)_=_(AutMt_(F1,b1,b1))_*_(i,j)
A11: Indices (AutMt ((id V1),b1,b1)) = Indices (AutMt (F1,b1,b1)) by MATRIX_1:26;
let i, j be Nat; ::_thesis: ( [i,j] in Indices (AutMt (F1,b1,b1)) implies (((AutMt (F1,b1,b1)) + ((- L) * (AutMt ((id V1),b1,b1)))) + (L * (AutMt ((id V1),b1,b1)))) * (i,j) = (AutMt (F1,b1,b1)) * (i,j) )
assume A12: [i,j] in Indices (AutMt (F1,b1,b1)) ; ::_thesis: (((AutMt (F1,b1,b1)) + ((- L) * (AutMt ((id V1),b1,b1)))) + (L * (AutMt ((id V1),b1,b1)))) * (i,j) = (AutMt (F1,b1,b1)) * (i,j)
Indices ((AutMt (F1,b1,b1)) + ((- L) * (AutMt ((id V1),b1,b1)))) = Indices (AutMt (F1,b1,b1)) by MATRIX_1:26;
hence (((AutMt (F1,b1,b1)) + ((- L) * (AutMt ((id V1),b1,b1)))) + (L * (AutMt ((id V1),b1,b1)))) * (i,j) = (((AutMt (F1,b1,b1)) + ((- L) * (AutMt ((id V1),b1,b1)))) * (i,j)) + ((L * (AutMt ((id V1),b1,b1))) * (i,j)) by A12, MATRIX_3:def_3
.= (((AutMt (F1,b1,b1)) * (i,j)) + (((- L) * (AutMt ((id V1),b1,b1))) * (i,j))) + ((L * (AutMt ((id V1),b1,b1))) * (i,j)) by A12, MATRIX_3:def_3
.= ((AutMt (F1,b1,b1)) * (i,j)) + ((((- L) * (AutMt ((id V1),b1,b1))) * (i,j)) + ((L * (AutMt ((id V1),b1,b1))) * (i,j))) by RLVECT_1:def_3
.= ((AutMt (F1,b1,b1)) * (i,j)) + (((- L) * ((AutMt ((id V1),b1,b1)) * (i,j))) + ((L * (AutMt ((id V1),b1,b1))) * (i,j))) by A12, A11, MATRIX_3:def_5
.= ((AutMt (F1,b1,b1)) * (i,j)) + (((- L) * ((AutMt ((id V1),b1,b1)) * (i,j))) + (L * ((AutMt ((id V1),b1,b1)) * (i,j)))) by A12, A11, MATRIX_3:def_5
.= ((AutMt (F1,b1,b1)) * (i,j)) + (((- L) + L) * ((AutMt ((id V1),b1,b1)) * (i,j))) by VECTSP_1:def_3
.= ((AutMt (F1,b1,b1)) * (i,j)) + ((0. K) * ((AutMt ((id V1),b1,b1)) * (i,j))) by VECTSP_1:16
.= ((AutMt (F1,b1,b1)) * (i,j)) + (0. K) by VECTSP_1:7
.= (AutMt (F1,b1,b1)) * (i,j) by RLVECT_1:def_4 ;
::_thesis: verum
end;
A13: Len (mlt (((len J) |-> L),(1. (K,(Len J))))) = Len (1. (K,(Len J))) by MATRIXJ1:62
.= Len J by MATRIXJ1:56 ;
dom (Len J) = dom (1. (K,(Len J))) by MATRIXJ1:def_8;
then A14: len (1. (K,(Len J))) = len (Len J) by FINSEQ_3:29
.= len J by CARD_1:def_7 ;
A15: Sum (Len J) = len (AutMt (FLV,b1,b1)) by A7, MATRIXJ1:def_5
.= len b1 by MATRIX_1:def_2 ;
A16: Width (mlt (((len J) |-> L),(1. (K,(Len J))))) = Width (1. (K,(Len J))) by MATRIXJ1:62
.= Len J by MATRIXJ1:56
.= Width J by MATRIXJ1:46 ;
Mx2Tran ((AutMt (((- L) * (id V1)),b1,b1)),b1,b1) = (- L) * (id V1) by MATRLIN2:34
.= (- L) * (Mx2Tran ((AutMt ((id V1),b1,b1)),b1,b1)) by MATRLIN2:34
.= Mx2Tran (((- L) * (AutMt ((id V1),b1,b1))),b1,b1) by MATRLIN2:38 ;
then AutMt (((- L) * (id V1)),b1,b1) = (- L) * (AutMt ((id V1),b1,b1)) by MATRLIN2:39;
then (AutMt (FLV,b1,b1)) + (L * (AutMt ((id V1),b1,b1))) = ((AutMt (F1,b1,b1)) + ((- L) * (AutMt ((id V1),b1,b1)))) + (L * (AutMt ((id V1),b1,b1))) by A6, MATRLIN:42
.= AutMt (F1,b1,b1) by A10, MATRIX_1:27 ;
hence AutMt (F1,b1,b1) = (block_diagonal (J,(0. K))) + (L * (1. (K,(Sum (Len J))))) by A7, A15, MATRLIN2:28
.= (block_diagonal (J,(0. K))) + (L * (block_diagonal ((1. (K,(Len J))),(0. K)))) by MATRIXJ1:61
.= (block_diagonal (J,(0. K))) + (block_diagonal ((mlt (((len J) |-> L),(1. (K,(Len J))))),(L * (0. K)))) by A14, MATRIXJ1:65
.= (block_diagonal (J,(0. K))) + (block_diagonal ((mlt (((len J) |-> L),(1. (K,(Len J))))),(0. K))) by VECTSP_1:7
.= block_diagonal (JM,((0. K) + (0. K))) by A13, A16, MATRIXJ1:72
.= block_diagonal (JM,(0. K)) by RLVECT_1:def_4 ;
::_thesis: verum
end;
begin
theorem Th31: :: MATRIXJ2:31
for K being algebraic-closed Field
for V being non trivial finite-dimensional VectSp of K
for F being linear-transformation of V,V ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) )
proof
let K be algebraic-closed Field; ::_thesis: for V being non trivial finite-dimensional VectSp of K
for F being linear-transformation of V,V ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) )
defpred S1[ Nat] means for V being non trivial finite-dimensional VectSp of K st dim V <= $1 holds
for F being linear-transformation of V,V ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) );
let V be non trivial finite-dimensional VectSp of K; ::_thesis: for F being linear-transformation of V,V ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) )
let F be linear-transformation of V,V; ::_thesis: ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) )
A1: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A2: S1[n] ; ::_thesis: S1[n + 1]
set n1 = n + 1;
let V be non trivial finite-dimensional VectSp of K; ::_thesis: ( dim V <= n + 1 implies for F being linear-transformation of V,V ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) ) )
assume A3: dim V <= n + 1 ; ::_thesis: for F being linear-transformation of V,V ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) )
percases ( dim V <= n or dim V = n + 1 ) by A3, NAT_1:8;
suppose dim V <= n ; ::_thesis: for F being linear-transformation of V,V ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) )
hence for F being linear-transformation of V,V ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) ) by A2; ::_thesis: verum
end;
supposeA4: dim V = n + 1 ; ::_thesis: for F being linear-transformation of V,V ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) )
let F be linear-transformation of V,V; ::_thesis: ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) )
A5: F is with_eigenvalues by VECTSP11:16;
then consider v being Vector of V, L being Scalar of K such that
A6: ( v <> 0. V & F . v = L * v ) by VECTSP11:def_1;
set FL = F + ((- L) * (id V));
L is eigenvalue of F by A5, A6, VECTSP11:def_2;
then not ker (F + ((- L) * (id V))) is trivial by A5, VECTSP11:14;
then A7: dim (ker (F + ((- L) * (id V)))) <> 0 by MATRLIN2:42;
consider m being Nat such that
A8: UnionKers (F + ((- L) * (id V))) = ker ((F + ((- L) * (id V))) |^ m) by VECTSP11:27;
set IM = im ((F + ((- L) * (id V))) |^ m);
set KER = ker ((F + ((- L) * (id V))) |^ m);
A9: dim V = (dim (ker ((F + ((- L) * (id V))) |^ m))) + (dim (im ((F + ((- L) * (id V))) |^ m))) by A8, VECTSP11:35, VECTSP_9:34;
A10: im ((F + ((- L) * (id V))) |^ m) is Linear_Compl of ker ((F + ((- L) * (id V))) |^ m) by A8, VECTSP11:35, VECTSP_5:37;
reconsider FK = F | (ker ((F + ((- L) * (id V))) |^ m)) as linear-transformation of (ker ((F + ((- L) * (id V))) |^ m)),(ker ((F + ((- L) * (id V))) |^ m)) by VECTSP11:29;
consider Jk being non-empty FinSequence_of_Jordan_block of L,K, Bker being OrdBasis of ker ((F + ((- L) * (id V))) |^ m) such that
A11: AutMt (FK,Bker,Bker) = block_diagonal (Jk,(0. K)) by Th30;
(F + ((- L) * (id V))) |^ 1 = F + ((- L) * (id V)) by VECTSP11:19;
then A12: ker (F + ((- L) * (id V))) is Subspace of ker ((F + ((- L) * (id V))) |^ m) by A8, VECTSP11:25;
A13: len Jk <> 0
proof
assume len Jk = 0 ; ::_thesis: contradiction
then Len Jk = <*> NAT
.= <*> REAL ;
then 0 = len (block_diagonal (Jk,(0. K))) by MATRIXJ1:def_5, RVSUM_1:72
.= len Bker by A11, MATRIX_1:def_2
.= dim (ker ((F + ((- L) * (id V))) |^ m)) by MATRLIN2:21 ;
hence contradiction by A12, A7, VECTSP_9:25; ::_thesis: verum
end;
reconsider FI = F | (im ((F + ((- L) * (id V))) |^ m)) as linear-transformation of (im ((F + ((- L) * (id V))) |^ m)),(im ((F + ((- L) * (id V))) |^ m)) by VECTSP11:33;
A14: (ker ((F + ((- L) * (id V))) |^ m)) /\ (im ((F + ((- L) * (id V))) |^ m)) = (0). V by A8, VECTSP11:34;
A15: V is_the_direct_sum_of ker ((F + ((- L) * (id V))) |^ m), im ((F + ((- L) * (id V))) |^ m) by A8, VECTSP11:35;
then A16: (ker ((F + ((- L) * (id V))) |^ m)) + (im ((F + ((- L) * (id V))) |^ m)) = (Omega). V by VECTSP_5:def_4;
percases ( im ((F + ((- L) * (id V))) |^ m) is trivial or not im ((F + ((- L) * (id V))) |^ m) is trivial ) ;
supposeA17: im ((F + ((- L) * (id V))) |^ m) is trivial ; ::_thesis: ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) )
set Bim = the OrdBasis of im ((F + ((- L) * (id V))) |^ m);
0 = dim (im ((F + ((- L) * (id V))) |^ m)) by A17, MATRLIN2:42
.= len the OrdBasis of im ((F + ((- L) * (id V))) |^ m) by MATRLIN2:21
.= len (AutMt (FI, the OrdBasis of im ((F + ((- L) * (id V))) |^ m), the OrdBasis of im ((F + ((- L) * (id V))) |^ m))) by MATRIX_1:def_2 ;
then A18: AutMt (FI, the OrdBasis of im ((F + ((- L) * (id V))) |^ m), the OrdBasis of im ((F + ((- L) * (id V))) |^ m)) = {} ;
Bker ^ the OrdBasis of im ((F + ((- L) * (id V))) |^ m) is OrdBasis of (ker ((F + ((- L) * (id V))) |^ m)) + (im ((F + ((- L) * (id V))) |^ m)) by A14, MATRLIN2:26;
then reconsider BB = Bker ^ the OrdBasis of im ((F + ((- L) * (id V))) |^ m) as OrdBasis of V by A16, MATRLIN2:4;
take Jk ; ::_thesis: ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (Jk,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom Jk & Jk . i = Jordan_block (L,(len (Jk . i))) ) ) ) )
take BB ; ::_thesis: ( AutMt (F,BB,BB) = block_diagonal (Jk,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom Jk & Jk . i = Jordan_block (L,(len (Jk . i))) ) ) ) )
A19: ( dim (im ((F + ((- L) * (id V))) |^ m)) = 0 implies dim (im ((F + ((- L) * (id V))) |^ m)) = 0 ) ;
( dim (ker ((F + ((- L) * (id V))) |^ m)) = 0 implies dim (ker ((F + ((- L) * (id V))) |^ m)) = 0 ) ;
hence AutMt (F,BB,BB) = block_diagonal (<*(AutMt (FK,Bker,Bker)),(AutMt (FI, the OrdBasis of im ((F + ((- L) * (id V))) |^ m), the OrdBasis of im ((F + ((- L) * (id V))) |^ m)))*>,(0. K)) by A15, A19, MATRLIN2:27
.= block_diagonal (<*(AutMt (FK,Bker,Bker))*>,(0. K)) by A18, MATRIXJ1:40
.= block_diagonal (Jk,(0. K)) by A11, MATRIXJ1:34 ;
::_thesis: for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom Jk & Jk . i = Jordan_block (L,(len (Jk . i))) ) )
let L1 be Scalar of K; ::_thesis: ( L1 is eigenvalue of F iff ex i being Nat st
( i in dom Jk & Jk . i = Jordan_block (L1,(len (Jk . i))) ) )
thus ( L1 is eigenvalue of F implies ex i being Nat st
( i in dom Jk & Jk . i = Jordan_block (L1,(len (Jk . i))) ) ) ::_thesis: ( ex i being Nat st
( i in dom Jk & Jk . i = Jordan_block (L1,(len (Jk . i))) ) implies L1 is eigenvalue of F )
proof
assume A20: L1 is eigenvalue of F ; ::_thesis: ex i being Nat st
( i in dom Jk & Jk . i = Jordan_block (L1,(len (Jk . i))) )
A21: L1 = L
proof
assume L <> L1 ; ::_thesis: contradiction
then ( FI is with_eigenvalues & L1 is eigenvalue of FI ) by A5, A8, A10, A20, VECTSP11:39;
then ex v1 being Vector of (im ((F + ((- L) * (id V))) |^ m)) st
( v1 <> 0. (im ((F + ((- L) * (id V))) |^ m)) & FI . v1 = L1 * v1 ) by VECTSP11:def_2;
hence contradiction by A17, STRUCT_0:def_18; ::_thesis: verum
end;
take i = len Jk; ::_thesis: ( i in dom Jk & Jk . i = Jordan_block (L1,(len (Jk . i))) )
i in Seg (len Jk) by A13, FINSEQ_1:3;
hence i in dom Jk by FINSEQ_1:def_3; ::_thesis: Jk . i = Jordan_block (L1,(len (Jk . i)))
then ex k being Nat st Jk . i = Jordan_block (L,k) by Def3;
hence Jk . i = Jordan_block (L1,(len (Jk . i))) by A21, Def1; ::_thesis: verum
end;
given i being Nat such that A22: i in dom Jk and
A23: Jk . i = Jordan_block (L1,(len (Jk . i))) ; ::_thesis: L1 is eigenvalue of F
Jk . i <> {} by A22, FUNCT_1:def_9;
then len (Jk . i) in Seg (len (Jk . i)) by FINSEQ_1:3;
then [(len (Jk . i)),(len (Jk . i))] in [:(Seg (len (Jk . i))),(Seg (len (Jk . i))):] by ZFMISC_1:87;
then A24: [(len (Jk . i)),(len (Jk . i))] in Indices (Jk . i) by MATRIX_1:24;
ex k being Nat st Jk . i = Jordan_block (L,k) by A22, Def3;
then L = (Jk . i) * ((len (Jk . i)),(len (Jk . i))) by A24, Def1
.= L1 by A23, A24, Def1 ;
hence L1 is eigenvalue of F by A5, A6, VECTSP11:def_2; ::_thesis: verum
end;
supposeA25: not im ((F + ((- L) * (id V))) |^ m) is trivial ; ::_thesis: ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) )
( n + 1 <> dim (im ((F + ((- L) * (id V))) |^ m)) & dim (im ((F + ((- L) * (id V))) |^ m)) <= n + 1 ) by A4, A12, A7, A9, VECTSP_9:25;
then dim (im ((F + ((- L) * (id V))) |^ m)) < n + 1 by XXREAL_0:1;
then dim (im ((F + ((- L) * (id V))) |^ m)) <= n by NAT_1:13;
then consider Ji being non-empty FinSequence_of_Jordan_block of K, Bim being OrdBasis of im ((F + ((- L) * (id V))) |^ m) such that
A26: AutMt (FI,Bim,Bim) = block_diagonal (Ji,(0. K)) and
A27: for L being Scalar of K holds
( L is eigenvalue of FI iff ex i being Nat st
( i in dom Ji & Ji . i = Jordan_block (L,(len (Ji . i))) ) ) by A2, A25;
Bker ^ Bim is OrdBasis of (ker ((F + ((- L) * (id V))) |^ m)) + (im ((F + ((- L) * (id V))) |^ m)) by A14, MATRLIN2:26;
then reconsider BB = Bker ^ Bim as OrdBasis of V by A16, MATRLIN2:4;
set JJ = Jk ^ Ji;
A28: now__::_thesis:_for_x_being_set_st_x_in_dom_(Jk_^_Ji)_holds_
not_(Jk_^_Ji)_._x_is_empty
let x be set ; ::_thesis: ( x in dom (Jk ^ Ji) implies not (Jk ^ Ji) . x is empty )
assume A29: x in dom (Jk ^ Ji) ; ::_thesis: not (Jk ^ Ji) . x is empty
reconsider i = x as Nat by A29;
now__::_thesis:_not_(Jk_^_Ji)_._i_is_empty
percases ( i in dom Jk or ex j being Nat st
( j in dom Ji & i = (len Jk) + j ) ) by A29, FINSEQ_1:25;
supposeA30: i in dom Jk ; ::_thesis: not (Jk ^ Ji) . i is empty
then (Jk ^ Ji) . i = Jk . i by FINSEQ_1:def_7;
hence not (Jk ^ Ji) . i is empty by A30, FUNCT_1:def_9; ::_thesis: verum
end;
suppose ex j being Nat st
( j in dom Ji & i = (len Jk) + j ) ; ::_thesis: not (Jk ^ Ji) . i is empty
then consider j being Nat such that
A31: j in dom Ji and
A32: i = (len Jk) + j ;
(Jk ^ Ji) . i = Ji . j by A31, A32, FINSEQ_1:def_7;
hence not (Jk ^ Ji) . i is empty by A31, FUNCT_1:def_9; ::_thesis: verum
end;
end;
end;
hence not (Jk ^ Ji) . x is empty ; ::_thesis: verum
end;
A33: FI is with_eigenvalues by A25, VECTSP11:16;
A34: ( dim (im ((F + ((- L) * (id V))) |^ m)) = 0 implies dim (im ((F + ((- L) * (id V))) |^ m)) = 0 ) ;
reconsider JJ = Jk ^ Ji as non-empty FinSequence_of_Jordan_block of K by A28, FUNCT_1:def_9;
take JJ ; ::_thesis: ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (JJ,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom JJ & JJ . i = Jordan_block (L,(len (JJ . i))) ) ) ) )
take BB ; ::_thesis: ( AutMt (F,BB,BB) = block_diagonal (JJ,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom JJ & JJ . i = Jordan_block (L,(len (JJ . i))) ) ) ) )
( dim (ker ((F + ((- L) * (id V))) |^ m)) = 0 implies dim (ker ((F + ((- L) * (id V))) |^ m)) = 0 ) ;
hence AutMt (F,BB,BB) = block_diagonal (<*(block_diagonal (Jk,(0. K))),(block_diagonal (Ji,(0. K)))*>,(0. K)) by A11, A15, A26, A34, MATRLIN2:27
.= block_diagonal ((<*(block_diagonal (Jk,(0. K)))*> ^ Ji),(0. K)) by MATRIXJ1:36
.= block_diagonal (JJ,(0. K)) by MATRIXJ1:35 ;
::_thesis: for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom JJ & JJ . i = Jordan_block (L,(len (JJ . i))) ) )
let L1 be Scalar of K; ::_thesis: ( L1 is eigenvalue of F iff ex i being Nat st
( i in dom JJ & JJ . i = Jordan_block (L1,(len (JJ . i))) ) )
thus ( L1 is eigenvalue of F implies ex i being Nat st
( i in dom JJ & JJ . i = Jordan_block (L1,(len (JJ . i))) ) ) ::_thesis: ( ex i being Nat st
( i in dom JJ & JJ . i = Jordan_block (L1,(len (JJ . i))) ) implies L1 is eigenvalue of F )
proof
assume A35: L1 is eigenvalue of F ; ::_thesis: ex i being Nat st
( i in dom JJ & JJ . i = Jordan_block (L1,(len (JJ . i))) )
percases ( L = L1 or L <> L1 ) ;
supposeA36: L = L1 ; ::_thesis: ex i being Nat st
( i in dom JJ & JJ . i = Jordan_block (L1,(len (JJ . i))) )
take i = len Jk; ::_thesis: ( i in dom JJ & JJ . i = Jordan_block (L1,(len (JJ . i))) )
A37: dom Jk c= dom JJ by FINSEQ_1:26;
i in Seg (len Jk) by A13, FINSEQ_1:3;
then A38: i in dom Jk by FINSEQ_1:def_3;
then ( ex k being Nat st Jk . i = Jordan_block (L,k) & JJ . i = Jk . i ) by Def3, FINSEQ_1:def_7;
hence ( i in dom JJ & JJ . i = Jordan_block (L1,(len (JJ . i))) ) by A36, A38, A37, Def1; ::_thesis: verum
end;
suppose L <> L1 ; ::_thesis: ex i being Nat st
( i in dom JJ & JJ . i = Jordan_block (L1,(len (JJ . i))) )
then L1 is eigenvalue of FI by A5, A8, A10, A35, VECTSP11:39;
then consider i being Nat such that
A39: i in dom Ji and
A40: Ji . i = Jordan_block (L1,(len (Ji . i))) by A27;
take ii = (len Jk) + i; ::_thesis: ( ii in dom JJ & JJ . ii = Jordan_block (L1,(len (JJ . ii))) )
JJ . ii = Ji . i by A39, FINSEQ_1:def_7;
hence ( ii in dom JJ & JJ . ii = Jordan_block (L1,(len (JJ . ii))) ) by A39, A40, FINSEQ_1:28; ::_thesis: verum
end;
end;
end;
given i being Nat such that A41: i in dom JJ and
A42: JJ . i = Jordan_block (L1,(len (JJ . i))) ; ::_thesis: L1 is eigenvalue of F
percases ( i in dom Jk or ex j being Nat st
( j in dom Ji & i = (len Jk) + j ) ) by A41, FINSEQ_1:25;
supposeA43: i in dom Jk ; ::_thesis: L1 is eigenvalue of F
then Jk . i <> {} by FUNCT_1:def_9;
then len (Jk . i) in Seg (len (Jk . i)) by FINSEQ_1:3;
then [(len (Jk . i)),(len (Jk . i))] in [:(Seg (len (Jk . i))),(Seg (len (Jk . i))):] by ZFMISC_1:87;
then A44: [(len (Jk . i)),(len (Jk . i))] in Indices (Jk . i) by MATRIX_1:24;
A45: JJ . i = Jk . i by A43, FINSEQ_1:def_7;
ex k being Nat st Jk . i = Jordan_block (L,k) by A43, Def3;
then L = (Jk . i) * ((len (Jk . i)),(len (Jk . i))) by A44, Def1
.= L1 by A42, A45, A44, Def1 ;
hence L1 is eigenvalue of F by A5, A6, VECTSP11:def_2; ::_thesis: verum
end;
suppose ex j being Nat st
( j in dom Ji & i = (len Jk) + j ) ; ::_thesis: L1 is eigenvalue of F
then consider j being Nat such that
A46: j in dom Ji and
A47: i = (len Jk) + j ;
JJ . i = Ji . j by A46, A47, FINSEQ_1:def_7;
then L1 is eigenvalue of FI by A27, A42, A46;
then consider w being Vector of (im ((F + ((- L) * (id V))) |^ m)) such that
A48: w <> 0. (im ((F + ((- L) * (id V))) |^ m)) and
A49: FI . w = L1 * w by A33, VECTSP11:def_2;
A50: 0. (im ((F + ((- L) * (id V))) |^ m)) = 0. V by VECTSP_4:11;
reconsider W = w as Vector of V by VECTSP_4:10;
L1 * W = FI . w by A49, VECTSP_4:14
.= F . W by FUNCT_1:49 ;
hence L1 is eigenvalue of F by A5, A48, A50, VECTSP11:def_2; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
A51: S1[ 0 ]
proof
let V be non trivial finite-dimensional VectSp of K; ::_thesis: ( dim V <= 0 implies for F being linear-transformation of V,V ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) ) )
assume dim V <= 0 ; ::_thesis: for F being linear-transformation of V,V ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) )
then dim V = 0 ;
hence for F being linear-transformation of V,V ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) ) by MATRLIN2:42; ::_thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch_2(A51, A1);
then S1[ dim V] ;
hence ex J being non-empty FinSequence_of_Jordan_block of K ex b1 being OrdBasis of V st
( AutMt (F,b1,b1) = block_diagonal (J,(0. K)) & ( for L being Scalar of K holds
( L is eigenvalue of F iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) ) ) ; ::_thesis: verum
end;
theorem :: MATRIXJ2:32
for n being Nat
for K being algebraic-closed Field
for M being Matrix of n,K ex J being non-empty FinSequence_of_Jordan_block of K ex P being Matrix of n,K st
( Sum (Len J) = n & P is invertible & M = (P * (block_diagonal (J,(0. K)))) * (P ~) )
proof
let n be Nat; ::_thesis: for K being algebraic-closed Field
for M being Matrix of n,K ex J being non-empty FinSequence_of_Jordan_block of K ex P being Matrix of n,K st
( Sum (Len J) = n & P is invertible & M = (P * (block_diagonal (J,(0. K)))) * (P ~) )
let K be algebraic-closed Field; ::_thesis: for M being Matrix of n,K ex J being non-empty FinSequence_of_Jordan_block of K ex P being Matrix of n,K st
( Sum (Len J) = n & P is invertible & M = (P * (block_diagonal (J,(0. K)))) * (P ~) )
let M be Matrix of n,K; ::_thesis: ex J being non-empty FinSequence_of_Jordan_block of K ex P being Matrix of n,K st
( Sum (Len J) = n & P is invertible & M = (P * (block_diagonal (J,(0. K)))) * (P ~) )
percases ( n = 0 or n > 0 ) ;
supposeA1: n = 0 ; ::_thesis: ex J being non-empty FinSequence_of_Jordan_block of K ex P being Matrix of n,K st
( Sum (Len J) = n & P is invertible & M = (P * (block_diagonal (J,(0. K)))) * (P ~) )
then reconsider P = {} as Matrix of n,K by MATRIX_1:13;
reconsider J = {} as FinSequence_of_Jordan_block of K by Lm2;
reconsider J = J as non-empty FinSequence_of_Jordan_block of K ;
take J ; ::_thesis: ex P being Matrix of n,K st
( Sum (Len J) = n & P is invertible & M = (P * (block_diagonal (J,(0. K)))) * (P ~) )
take P ; ::_thesis: ( Sum (Len J) = n & P is invertible & M = (P * (block_diagonal (J,(0. K)))) * (P ~) )
A2: Len J = <*> NAT by CARD_1:27
.= <*> REAL ;
hence Sum (Len J) = n by A1, RVSUM_1:72; ::_thesis: ( P is invertible & M = (P * (block_diagonal (J,(0. K)))) * (P ~) )
A3: 1_ K <> 0. K ;
Det P = 1_ K by A1, MATRIXR2:41;
hence P is invertible by A3, LAPLACE:34; ::_thesis: M = (P * (block_diagonal (J,(0. K)))) * (P ~)
reconsider B = block_diagonal (J,(0. K)) as Matrix of n,K by A1, A2, RVSUM_1:72;
M = (P * B) * (P ~) by A1, MATRIX_1:35;
hence M = (P * (block_diagonal (J,(0. K)))) * (P ~) ; ::_thesis: verum
end;
supposeA4: n > 0 ; ::_thesis: ex J being non-empty FinSequence_of_Jordan_block of K ex P being Matrix of n,K st
( Sum (Len J) = n & P is invertible & M = (P * (block_diagonal (J,(0. K)))) * (P ~) )
set V = n -VectSp_over K;
set B = the OrdBasis of n -VectSp_over K;
A5: len the OrdBasis of n -VectSp_over K = dim (n -VectSp_over K) by MATRLIN2:21
.= n by MATRIX13:112 ;
then reconsider M9 = M as Matrix of len the OrdBasis of n -VectSp_over K,K ;
set T = Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K);
dim (n -VectSp_over K) = n by MATRIX13:112;
then not n -VectSp_over K is trivial by A4, MATRLIN2:42;
then consider J being non-empty FinSequence_of_Jordan_block of K, b1 being OrdBasis of n -VectSp_over K such that
A6: AutMt ((Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K)),b1,b1) = block_diagonal (J,(0. K)) and
for L being Scalar of K holds
( L is eigenvalue of Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K) iff ex i being Nat st
( i in dom J & J . i = Jordan_block (L,(len (J . i))) ) ) by Th31;
A7: dom (Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K)) = [#] (n -VectSp_over K) by FUNCT_2:def_1;
reconsider P = AutEqMt ((id (n -VectSp_over K)), the OrdBasis of n -VectSp_over K,b1) as Matrix of n,K by A5;
take J ; ::_thesis: ex P being Matrix of n,K st
( Sum (Len J) = n & P is invertible & M = (P * (block_diagonal (J,(0. K)))) * (P ~) )
take P ; ::_thesis: ( Sum (Len J) = n & P is invertible & M = (P * (block_diagonal (J,(0. K)))) * (P ~) )
A8: ( width P = n & len (P ~) = n ) by A4, MATRIX_1:23;
A9: rng (Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K)) c= [#] (n -VectSp_over K) by RELAT_1:def_19;
A10: len b1 = dim (n -VectSp_over K) by MATRLIN2:21
.= n by MATRIX13:112 ;
then A11: ( len (AutMt ((Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K)),b1,b1)) = n & width (AutMt ((Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K)),b1,b1)) = n ) by A4, MATRIX_1:23;
thus Sum (Len J) = len (AutMt ((Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K)),b1,b1)) by A6, MATRIXJ1:def_5
.= n by A10, MATRIX_1:def_2 ; ::_thesis: ( P is invertible & M = (P * (block_diagonal (J,(0. K)))) * (P ~) )
thus P is invertible by A5, MATRLIN2:29; ::_thesis: M = (P * (block_diagonal (J,(0. K)))) * (P ~)
thus M = AutMt ((Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K)), the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K) by MATRLIN2:36
.= AutMt (((Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K)) * (id (n -VectSp_over K))), the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K) by A7, RELAT_1:52
.= (AutMt ((id (n -VectSp_over K)), the OrdBasis of n -VectSp_over K,b1)) * (AutMt ((Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K)),b1, the OrdBasis of n -VectSp_over K)) by A4, A5, A10, MATRLIN:41
.= P * (AutMt ((Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K)),b1, the OrdBasis of n -VectSp_over K)) by A5, A10, MATRLIN2:def_2
.= P * (AutMt (((id (n -VectSp_over K)) * (Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K))),b1, the OrdBasis of n -VectSp_over K)) by A9, RELAT_1:53
.= P * ((AutMt ((Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K)),b1,b1)) * (AutMt ((id (n -VectSp_over K)),b1, the OrdBasis of n -VectSp_over K))) by A4, A10, MATRLIN:41
.= P * ((AutMt ((Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K)),b1,b1)) * (AutEqMt ((id (n -VectSp_over K)),b1, the OrdBasis of n -VectSp_over K))) by A5, A10, MATRLIN2:def_2
.= P * ((AutMt ((Mx2Tran (M9, the OrdBasis of n -VectSp_over K, the OrdBasis of n -VectSp_over K)),b1,b1)) * (P ~)) by A5, MATRLIN2:29
.= (P * (block_diagonal (J,(0. K)))) * (P ~) by A6, A8, A11, MATRIX_3:33 ; ::_thesis: verum
end;
end;
end;