:: The Product of Matrices of Elements of a Field and Determinants
:: by Katarzyna Zawadzka
::
:: Received May 17, 1993
:: Copyright (c) 1993-2021 Association of Mizar Users

definition

let K be Ring;
let n, m be Nat;

func 0. (K,n,m) -> Matrix of n,m,K equals :: MATRIX_3:def 1
n |-> (m |-> (0. K));

coherence
n |-> (m |-> (0. K)) is Matrix of n,m,K by MATRIX_0:10;

end;

:: deftheorem defines 0. MATRIX_3:def 1 :
for K being Ring
for n, m being Nat holds 0. (K,n,m) = n |-> (m |-> (0. K));

definition

let K be Ring;
let A be Matrix of K;

func - A -> Matrix of K means :Def2: :: MATRIX_3:def 2
( len it = len A & width it = width A & ( for i, j being Nat st [i,j] in Indices A holds
it * (i,j) = - (A * (i,j)) ) );

existence
ex b₁ being Matrix of K st
( len b₁ = len A & width b₁ = width A & ( for i, j being Nat st [i,j] in Indices A holds
b₁ * (i,j) = - (A * (i,j)) ) )

proof end;

uniqueness
for b₁, b₂ being Matrix of K st len b₁ = len A & width b₁ = width A & ( for i, j being Nat st [i,j] in Indices A holds
b₁ * (i,j) = - (A * (i,j)) ) & len b₂ = len A & width b₂ = width A & ( for i, j being Nat st [i,j] in Indices A holds
b₂ * (i,j) = - (A * (i,j)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines - MATRIX_3:def 2 :
for K being Ring
for A, b₃ being Matrix of K holds
( b₃ = - A iff ( len b₃ = len A & width b₃ = width A & ( for i, j being Nat st [i,j] in Indices A holds
b₃ * (i,j) = - (A * (i,j)) ) ) );

definition

let K be Ring;
let A, B be Matrix of K;

func A + B -> Matrix of K means :Def3: :: MATRIX_3:def 3
( len it = len A & width it = width A & ( for i, j being Nat st [i,j] in Indices A holds
it * (i,j) = (A * (i,j)) + (B * (i,j)) ) );

existence
ex b₁ being Matrix of K st
( len b₁ = len A & width b₁ = width A & ( for i, j being Nat st [i,j] in Indices A holds
b₁ * (i,j) = (A * (i,j)) + (B * (i,j)) ) )

proof end;

uniqueness
for b₁, b₂ being Matrix of K st len b₁ = len A & width b₁ = width A & ( for i, j being Nat st [i,j] in Indices A holds
b₁ * (i,j) = (A * (i,j)) + (B * (i,j)) ) & len b₂ = len A & width b₂ = width A & ( for i, j being Nat st [i,j] in Indices A holds
b₂ * (i,j) = (A * (i,j)) + (B * (i,j)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines + MATRIX_3:def 3 :
for K being Ring
for A, B, b₄ being Matrix of K holds
( b₄ = A + B iff ( len b₄ = len A & width b₄ = width A & ( for i, j being Nat st [i,j] in Indices A holds
b₄ * (i,j) = (A * (i,j)) + (B * (i,j)) ) ) );

theorem Th1: :: MATRIX_3:1

for n, m being Nat
for K being Ring
for i, j being Nat st [i,j] in Indices (0. (K,n,m)) holds
(0. (K,n,m)) * (i,j) = 0. K

proof end;

theorem :: MATRIX_3:2

for K being Ring
for A, B being Matrix of K st len A = len B & width A = width B holds
A + B = B + A

proof end;

theorem :: MATRIX_3:3

for K being Ring
for A, B, C being Matrix of K st len A = len B & width A = width B holds
(A + B) + C = A + (B + C)

proof end;

theorem :: MATRIX_3:4

for n, m being Nat
for K being Ring
for A being Matrix of n,m,K holds A + (0. (K,n,m)) = A

proof end;

theorem :: MATRIX_3:5

for n, m being Nat
for K being Ring
for A being Matrix of n,m,K holds A + (- A) = 0. (K,n,m)

proof end;

definition

let K be Ring;
let A, B be Matrix of K;
assume A1: width A = len B ;

func A * B -> Matrix of K means :Def4: :: MATRIX_3:def 4
( len it = len A & width it = width B & ( for i, j being Nat st [i,j] in Indices it holds
it * (i,j) = (Line (A,i)) "*" (Col (B,j)) ) );

existence
ex b₁ being Matrix of K st
( len b₁ = len A & width b₁ = width B & ( for i, j being Nat st [i,j] in Indices b₁ holds
b₁ * (i,j) = (Line (A,i)) "*" (Col (B,j)) ) )

proof end;

uniqueness
for b₁, b₂ being Matrix of K st len b₁ = len A & width b₁ = width B & ( for i, j being Nat st [i,j] in Indices b₁ holds
b₁ * (i,j) = (Line (A,i)) "*" (Col (B,j)) ) & len b₂ = len A & width b₂ = width B & ( for i, j being Nat st [i,j] in Indices b₂ holds
b₂ * (i,j) = (Line (A,i)) "*" (Col (B,j)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines * MATRIX_3:def 4 :
for K being Ring
for A, B being Matrix of K st width A = len B holds
for b₄ being Matrix of K holds
( b₄ = A * B iff ( len b₄ = len A & width b₄ = width B & ( for i, j being Nat st [i,j] in Indices b₄ holds
b₄ * (i,j) = (Line (A,i)) "*" (Col (B,j)) ) ) );

definition

let n, k, m be Nat;
let K be Ring;
let A be Matrix of n,k,K;
let B be Matrix of width A,m,K;
:: original: *
redefine func A * B -> Matrix of len A, width B,K;
coherence
A * B is Matrix of len A, width B,K

proof end;

end;

definition

let K be Ring;
let M be Matrix of K;
let a be Element of K;

func a * M -> Matrix of K means :: MATRIX_3:def 5
( len it = len M & width it = width M & ( for i, j being Nat st [i,j] in Indices M holds
it * (i,j) = a * (M * (i,j)) ) );

existence
ex b₁ being Matrix of K st
( len b₁ = len M & width b₁ = width M & ( for i, j being Nat st [i,j] in Indices M holds
b₁ * (i,j) = a * (M * (i,j)) ) )

proof end;

uniqueness
for b₁, b₂ being Matrix of K st len b₁ = len M & width b₁ = width M & ( for i, j being Nat st [i,j] in Indices M holds
b₁ * (i,j) = a * (M * (i,j)) ) & len b₂ = len M & width b₂ = width M & ( for i, j being Nat st [i,j] in Indices M holds
b₂ * (i,j) = a * (M * (i,j)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem defines * MATRIX_3:def 5 :
for K being Ring
for M being Matrix of K
for a being Element of K
for b₄ being Matrix of K holds
( b₄ = a * M iff ( len b₄ = len M & width b₄ = width M & ( for i, j being Nat st [i,j] in Indices M holds
b₄ * (i,j) = a * (M * (i,j)) ) ) );

definition

let K be Ring;
let M be Matrix of K;
let a be Element of K;

func M * a -> Matrix of K equals :: MATRIX_3:def 6
a * M;

coherence
a * M is Matrix of K ;

end;

:: deftheorem defines * MATRIX_3:def 6 :
for K being Ring
for M being Matrix of K
for a being Element of K holds M * a = a * M;

theorem :: MATRIX_3:6

for K being Ring
for p, q being FinSequence of K st len p = len q holds
( len (mlt (p,q)) = len p & len (mlt (p,q)) = len q )

proof end;

theorem :: MATRIX_3:7

for n being Nat
for K being Ring
for i, l being Nat st [i,l] in Indices (1. (K,n)) & l = i holds
(Line ((1. (K,n)),i)) . l = 1. K

proof end;

theorem :: MATRIX_3:8

for n being Nat
for K being Ring
for i, l being Nat st [i,l] in Indices (1. (K,n)) & l <> i holds
(Line ((1. (K,n)),i)) . l = 0. K

proof end;

theorem :: MATRIX_3:9

for n being Nat
for K being Ring
for l, j being Nat st [l,j] in Indices (1. (K,n)) & l = j holds
(Col ((1. (K,n)),j)) . l = 1. K

proof end;

theorem :: MATRIX_3:10

for n being Nat
for K being Ring
for l, j being Nat st [l,j] in Indices (1. (K,n)) & l <> j holds
(Col ((1. (K,n)),j)) . l = 0. K

proof end;

Lm1: for L being non empty right_complementable add-associative right_zeroed addLoopStr
for p being FinSequence of L st ( for i being Element of NAT st i in dom p holds
p . i = 0. L ) holds
Sum p = 0. L

proof end;

theorem Th11: :: MATRIX_3:11

for n being Element of NAT
for K being non empty right_complementable add-associative right_zeroed addLoopStr holds Sum (n |-> (0. K)) = 0. K

proof end;

theorem Th12: :: MATRIX_3:12

for K being non empty right_complementable add-associative right_zeroed addLoopStr
for p being FinSequence of K
for i being Nat st i in dom p & ( for k being Nat st k in dom p & k <> i holds
p . k = 0. K ) holds
Sum p = p . i

proof end;

theorem Th13: :: MATRIX_3:13

for K being Ring
for p, q being FinSequence of K holds len (mlt (p,q)) = min ((len p),(len q))

proof end;

theorem Th14: :: MATRIX_3:14

for K being Ring
for p, q being FinSequence of K
for i being Nat st p . i = 1. K & ( for k being Nat st k in dom p & k <> i holds
p . k = 0. K ) holds
for j being Nat st j in dom (mlt (p,q)) holds
( ( i = j implies (mlt (p,q)) . j = q . i ) & ( i <> j implies (mlt (p,q)) . j = 0. K ) )

proof end;

theorem Th15: :: MATRIX_3:15

for n being Nat
for K being Ring
for i, j being Nat st [i,j] in Indices (1. (K,n)) holds
( ( i = j implies (Line ((1. (K,n)),i)) . j = 1. K ) & ( i <> j implies (Line ((1. (K,n)),i)) . j = 0. K ) )

proof end;

theorem Th16: :: MATRIX_3:16

for n being Nat
for K being Ring
for i, j being Nat st [i,j] in Indices (1. (K,n)) holds
( ( i = j implies (Col ((1. (K,n)),j)) . i = 1. K ) & ( i <> j implies (Col ((1. (K,n)),j)) . i = 0. K ) )

proof end;

theorem Th17: :: MATRIX_3:17

for K being Ring
for p, q being FinSequence of K
for i being Nat st i in dom p & i in dom q & p . i = 1. K & ( for k being Nat st k in dom p & k <> i holds
p . k = 0. K ) holds
Sum (mlt (p,q)) = q . i

proof end;

theorem :: MATRIX_3:18

for n being Nat
for K being Ring
for A being Matrix of n,K holds (1. (K,n)) * A = A

proof end;

theorem :: MATRIX_3:19

for n being Nat
for K being commutative Ring
for A being Matrix of n,K holds A * (1. (K,n)) = A

proof end;

theorem :: MATRIX_3:20

for K being Field
for a, b being Element of K holds <*<*a*>*> * <*<*b*>*> = <*<*(a * b)*>*>

proof end;

theorem :: MATRIX_3:21

for K being Field
for a1, a2, b1, b2, c1, c2, d1, d2 being Element of K holds ((a1,b1) ][ (c1,d1)) * ((a2,b2) ][ (c2,d2)) = (((a1 * a2) + (b1 * c2)),((a1 * b2) + (b1 * d2))) ][ (((c1 * a2) + (d1 * c2)),((c1 * b2) + (d1 * d2)))

proof end;

theorem :: MATRIX_3:22

for K being Field
for A, B being Matrix of K st width A = len B & width B <> 0 holds
(A * B) @ = (B @) * (A @)

proof end;

definition

let I, J be non empty set ;
let X be Element of Fin I;
let Y be Element of Fin J;
:: original: [:
redefine func [:X,Y:] -> Element of Fin [:I,J:];
coherence
[:X,Y:] is Element of Fin [:I,J:]

proof end;

end;

definition

let I, J, D be non empty set ;
let G be BinOp of D;
let f be Function of I,D;
let g be Function of J,D;
:: original: *
redefine func G * (f,g) -> Function of [:I,J:],D;
coherence
G * (f,g) is Function of [:I,J:],D by FINSEQOP:79;

end;

theorem Th23: :: MATRIX_3:23

for I, J, D being non empty set
for F, G being BinOp of D
for f being Function of I,D
for g being Function of J,D
for X being Element of Fin I
for Y being Element of Fin J st F is commutative & F is associative & ( [:Y,X:] <> {} or F is having_a_unity ) & G is commutative holds
F $$ ([:X,Y:],(G * (f,g))) = F $$ ([:Y,X:],(G * (g,f)))

proof end;

theorem Th24: :: MATRIX_3:24

for D, I, J being non empty set
for F, G being BinOp of D
for f being Function of I,D
for g being Function of J,D st F is commutative & F is associative holds
for x being Element of I
for y being Element of J holds F $$ ([:{.x.},{.y.}:],(G * (f,g))) = F $$ ({.x.},(G [:] (f,(F $$ ({.y.},g)))))

proof end;

theorem Th25: :: MATRIX_3:25

for D, I, J being non empty set
for F, G being BinOp of D
for f being Function of I,D
for g being Function of J,D
for X being Element of Fin I
for Y being Element of Fin J st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds
for x being Element of I holds F $$ ([:{.x.},Y:],(G * (f,g))) = F $$ ({.x.},(G [:] (f,(F $$ (Y,g)))))

proof end;

theorem Th26: :: MATRIX_3:26

for D, I, J being non empty set
for F, G being BinOp of D
for f being Function of I,D
for g being Function of J,D
for X being Element of Fin I
for Y being Element of Fin J st F is having_a_unity & F is commutative & F is associative & F is having_an_inverseOp & G is_distributive_wrt F holds
F $$ ([:X,Y:],(G * (f,g))) = F $$ (X,(G [:] (f,(F $$ (Y,g)))))

proof end;

theorem :: MATRIX_3:27

for D, I, J being non empty set
for F, G being BinOp of D
for f being Function of I,D
for g being Function of J,D st F is commutative & F is associative & G is commutative holds
for x being Element of I
for y being Element of J holds F $$ ([:{.x.},{.y.}:],(G * (f,g))) = F $$ ({.y.},(G [;] ((F $$ ({.x.},f)),g)))

proof end;

theorem :: MATRIX_3:28

for D, I, J being non empty set
for F, G being BinOp of D
for f being Function of I,D
for g being Function of J,D
for X being Element of Fin I
for Y being Element of Fin J st F is having_a_unity & F is commutative & F is associative & F is having_an_inverseOp & G is_distributive_wrt F & G is commutative holds
F $$ ([:X,Y:],(G * (f,g))) = F $$ (Y,(G [;] ((F $$ (X,f)),g)))

proof end;

theorem Th29: :: MATRIX_3:29

for D, I, J being non empty set
for F being BinOp of D
for f being Function of [:I,J:],D
for g being Function of I,D
for Y being Element of Fin J st F is having_a_unity & F is commutative & F is associative holds
for x being Element of I st ( for i being Element of I holds g . i = F $$ (Y,((curry f) . i)) ) holds
F $$ ([:{.x.},Y:],f) = F $$ ({.x.},g)

proof end;

theorem Th30: :: MATRIX_3:30

for D, I, J being non empty set
for F being BinOp of D
for f being Function of [:I,J:],D
for g being Function of I,D
for X being Element of Fin I
for Y being Element of Fin J st ( for i being Element of I holds g . i = F $$ (Y,((curry f) . i)) ) & F is having_a_unity & F is commutative & F is associative holds
F $$ ([:X,Y:],f) = F $$ (X,g)

proof end;

theorem Th31: :: MATRIX_3:31

for D, I, J being non empty set
for F being BinOp of D
for f being Function of [:I,J:],D
for g being Function of J,D
for X being Element of Fin I st F is having_a_unity & F is commutative & F is associative holds
for y being Element of J st ( for j being Element of J holds g . j = F $$ (X,((curry' f) . j)) ) holds
F $$ ([:X,{.y.}:],f) = F $$ ({.y.},g)

proof end;

theorem Th32: :: MATRIX_3:32

for D, I, J being non empty set
for F being BinOp of D
for f being Function of [:I,J:],D
for g being Function of J,D
for X being Element of Fin I
for Y being Element of Fin J st ( for j being Element of J holds g . j = F $$ (X,((curry' f) . j)) ) & F is having_a_unity & F is commutative & F is associative holds
F $$ ([:X,Y:],f) = F $$ (Y,g)

proof end;

theorem :: MATRIX_3:33

for K being commutative Ring
for A, B, C being Matrix of K st width A = len B & width B = len C holds
(A * B) * C = A * (B * C)

proof end;

definition

let n be Nat;
let K be Ring;
let M be Matrix of n,K;
let p be Element of Permutations n;

func Path_matrix (p,M) -> FinSequence of K means :Def7: :: MATRIX_3:def 7
( len it = n & ( for i, j being Nat st i in dom it & j = p . i holds
it . i = M * (i,j) ) );

existence
ex b₁ being FinSequence of K st
( len b₁ = n & ( for i, j being Nat st i in dom b₁ & j = p . i holds
b₁ . i = M * (i,j) ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of K st len b₁ = n & ( for i, j being Nat st i in dom b₁ & j = p . i holds
b₁ . i = M * (i,j) ) & len b₂ = n & ( for i, j being Nat st i in dom b₂ & j = p . i holds
b₂ . i = M * (i,j) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def7 defines Path_matrix MATRIX_3:def 7 :
for n being Nat
for K being Ring
for M being Matrix of n,K
for p being Element of Permutations n
for b₅ being FinSequence of K holds
( b₅ = Path_matrix (p,M) iff ( len b₅ = n & ( for i, j being Nat st i in dom b₅ & j = p . i holds
b₅ . i = M * (i,j) ) ) );

definition

let n be Nat;
let K be Ring;
let M be Matrix of n,K;

func Path_product M -> Function of (Permutations n), the carrier of K means :Def8: :: MATRIX_3:def 8
for p being Element of Permutations n holds it . p = - (( the multF of K $$ (Path_matrix (p,M))),p);

existence
ex b₁ being Function of (Permutations n), the carrier of K st
for p being Element of Permutations n holds b₁ . p = - (( the multF of K $$ (Path_matrix (p,M))),p)

proof end;

uniqueness
for b₁, b₂ being Function of (Permutations n), the carrier of K st ( for p being Element of Permutations n holds b₁ . p = - (( the multF of K $$ (Path_matrix (p,M))),p) ) & ( for p being Element of Permutations n holds b₂ . p = - (( the multF of K $$ (Path_matrix (p,M))),p) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def8 defines Path_product MATRIX_3:def 8 :
for n being Nat
for K being Ring
for M being Matrix of n,K
for b₄ being Function of (Permutations n), the carrier of K holds
( b₄ = Path_product M iff for p being Element of Permutations n holds b₄ . p = - (( the multF of K $$ (Path_matrix (p,M))),p) );

definition

let n be Nat;
let K be Ring;
let M be Matrix of n,K;

func Det M -> Element of K equals :: MATRIX_3:def 9
the addF of K $$ ((In ((Permutations n),(Fin (Permutations n)))),(Path_product M));

coherence
the addF of K $$ ((In ((Permutations n),(Fin (Permutations n)))),(Path_product M)) is Element of K ;

end;

:: deftheorem defines Det MATRIX_3:def 9 :
for n being Nat
for K being Ring
for M being Matrix of n,K holds Det M = the addF of K $$ ((In ((Permutations n),(Fin (Permutations n)))),(Path_product M));

theorem :: MATRIX_3:34

for K being Ring
for a being Element of K holds Det <*<*a*>*> = a

proof end;

definition

let n be Nat;
let K be Ring;
let M be Matrix of n,K;

func diagonal_of_Matrix M -> FinSequence of K means :: MATRIX_3:def 10
( len it = n & ( for i being Nat st i in Seg n holds
it . i = M * (i,i) ) );

existence
ex b₁ being FinSequence of K st
( len b₁ = n & ( for i being Nat st i in Seg n holds
b₁ . i = M * (i,i) ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of K st len b₁ = n & ( for i being Nat st i in Seg n holds
b₁ . i = M * (i,i) ) & len b₂ = n & ( for i being Nat st i in Seg n holds
b₂ . i = M * (i,i) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem defines diagonal_of_Matrix MATRIX_3:def 10 :
for n being Nat
for K being Ring
for M being Matrix of n,K
for b₄ being FinSequence of K holds
( b₄ = diagonal_of_Matrix M iff ( len b₄ = n & ( for i being Nat st i in Seg n holds
b₄ . i = M * (i,i) ) ) );