(,) A (,) B C = AB a 11 a 1 a 1 b 11 b 1 b 1 c 11 c 1 c a A = 1 a a, B = b 1 b b, C = AB = c 1 c c a 1 a a b 1 b b c 1 c c i j ij a i1 a i a i b 1j b j b j c ij = a ik b kj
b 1j b j AB = a i1 a i a ik a i = b kj a ik b kj = C b j (,) AO = O A = O AE = A, E A = A O O E 1 0 1 1 0 0 1 A = 0 1 1 0, B = 0 1 0 1 0 1 1 0 0 1 1 1 0 AB BA AB BA 0 0 0 0 1 1 AB = 1 1 1, BA = 0 1 1 0 0 0 1 1 1 1 1 1
i j a ij = 1 a ij = 0 a ij (i, j) A A 0 1 0 1 0 0 1 0 A = 0 1 0 1 1 1 1 0 0 1 0 1 0 1 0 1 1 1 0 A 0 0 1 0 = 0 0 1 0 = 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 1 1 0 0 1 1.4.1 1.4. 1..3 1.4.3.3..3.4 3.4.1 3.4. 3..3 3.4.3 4.1. 4.1.4 4..3 4.3. 4.3.4 A (i, j) (i, j) A A 3 A 3 a i1 a 1j + a i a j + a i3 a 3j + a i4 a 4 j i j i j 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 3 1 3 A 3 0 1 0 1 = 0 0 1 0 = 0 0 1 0 0 1 0 1 = 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 3 1 3 0 1 1 1 1 0 1 1 1 0 0 1 3 4 1 4 1 3 4 1 4 3 4 3 3 4 3 4 3
A A 0 0 0 0 1 1 0 1 1 0 A = 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 3 A = 1 0 0 0 1 1 0 0 0 1 = 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 k 1 5 3 5 4 5 a 11 a 1 a 1 a A = 1 a a a 1 a a 1 = a 1t = a 11 + a 1 ++ a 1 t =1 = a t = a 1 + a ++ a t =1 = a t = a 1 + a ++ a t =1 1 + ++
s = t t =1 = A 1 = a s1 = a 11 + a 1 ++ a 1 = a s = a 1 + a ++ a = a s = a 1 + a ++ a 1 + ++ a st = a st t =1 t =1 = A 1 3 +1 + + 3 A = +1 + + 3 3 ( 1) +1 ( 1) + ( 1) + 3 s = a st s t =1 t = a st t t =1 s = t t =1
a st = (s 1) + t s = a st = ((s 1) + t) = (s 1) + t =1 t =1 s = (s 1) + ( +1) ) = 3 ( 1) ( * t = a st = ((s 1) + t) = ( 1) + t * t = t =1 t =1 ( 1) + t) = 3 ( 1) ( + s = * t t =1 ( +1) + ( +1) + ( +1) = ( + +1) = 4 3 + 3 + ) = ( +1) = ( +1) C (,) (,) (A + B)C = AC + BC C (,) (,) A(B + C) = AB + BC A = [ aij ], B = [ b ij ], A + B = [ ij ] = [ a ij + b ij ], C = [ c ij ] {(A + B)C} ij = ik c kj = (a ik + b ik )c kj = (a ik c kj + b ik c kj ) = a ik c kj + b ik c kj = AC (A + B)C = AC + BC { } ij + { BC} ij A(B + C) = AB + BC
(,) (,) (,p) (AB)C = A(BC) A = a ij, B = b ij, AB = ij = a b is sj ( ), {(AB)C} ij = it c tj = a is b * + st -. c ), tj = a is b st c * + tj -. t=1 t=1 t=1 ), ), = a is b st c * + tj -. = a is b st c * + tj -. t=1 t=1 = { A(BC) } ij A = [ x 1 x x 3 ] A = x y c [ ] a c y 1 y y 3 0 a b A = 0 0 a 0 0 0 0 a b x A = 0 0 a b y 0 0 0 A = x 1 y 1 + x y + x 3 y 3 A = ax + cxy + by 0 0 a A = 0 0 0 0 0 0 0 0 0 A = 0 0 0 = O 0 0 0 3
E A = (E A)(E + A) E A 3 = (E A)(E + A + A ) (E A)(E + A) = E(E + A) A(E + A) = E + EA AE A = E + A A A = E A (E A)(E + A + A ) = E(E + A + A ) A(E + A + A ) = E + EA + EA AE A A 3 = E + A + A A A A 3 = E A 3 A + B = B + A AB BA (,) E Kroecker s delta ij = 1 0 i = j i j 1 0 0 0 1 0 E = [ ij ] = 0 0 1
A E A = [ a ij ], E = ij AE = EA = ) AE = EA = A [ ] a ik kj ( ( = a i1 1j + a i j +a i j [ ] = [ a ij ] = A ik a kj ( ( = i1 a 1j + i a j + i a j [ ] = [ a ij ] = A X 1 0 X = 0 1 0 0 X 0 1 0 1 0 A = 0 0 1 X = 0 1 0 0 0 0 0 = (E + A) 0 0 1 0 0 0 A = 0 0 0, A 3 = 0 0 0 = O, A = O for ( 3 0 0 0 0 0 0 x, y (x + y) = x k y (k, k k= 0 = k C k = ( ( k)k A,B (A + B) = A k B (k, k k= 0 = k C k = ( ( k)k
E A X = (E + A) = ) (E) (k A k = E + (1 A + 1 k ( (1) ( A + ca 3 + k= 0 = E + (1 A + 1 ( (1) ( A * (1 1 (-, ( (1) /, = 0 (1 /, /, 0 0 / +,./ 0 1 0 0 0 0 0 1 0 A = 0 0 1, B = 0 1 0, X = A + B = 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 AB = 0 0 0, BA = 0 0, A = 0 0 0, B = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 X = A + AB + BA + B = 0, X = A + AB + B = 0 0 0 0 0 0 0 0 1 0 1 3 3 3 4 4 3 6 X = 0 1, X = 0, X 3 = 0 3 3, X 4 = 0 4 4 3 0 0 0 0 0 0 3 0 0 4 (1 1 ( ( (1) X = 0 (1 0 0 1 1 ( 1) ( 1 0 +1 ( +1) 1 1 ( +1) ( X +1 = X X = 0 1 ( ( ( 0 1 ( = 0 +1 ( +1) ( ( 0 0 ( 0 0 ( 0 0 +1 ( ( (
X = X = 0 a 1 a 13 a 1 0 0 a 3 a 0 0 0 a (1) 0 0 0 0 0 a 1 a 13 a 1 0 0 a 3 a 0 0 0 a (1) 0 0 0 0 = 0 0 * * * 0 0 0 * * 0 0 * 0 0 0 0 = = 1 X +1 = X 1 X = 0 a 1 a 13 a 1 0 0 a 3 a 0 0 0 a (1) 0 0 0 0 0 0 * * * 0 0 0 * * 0 0 * 0 0 0 0 0 0 * * * 0 0 0 * * X +1 = X 1 X = 0 0 * 0 0 0 0 + 1 = X = X = O
A t A trasposed a 11 a 1 a 1 a A = 1 a a a 1 a a a 11 a 1 a 1 t a A = 1 a a a 1 a a (,) t ( t A) = A t (A + B)= t A+ t B t (A) = t A
(,) (,) t (AB)= t B t A A = [ a ij ], B = [ b ij ], a ij = A { } ij, b ij = { B} ij { t (AB)} = a jk b ki ij = b ki a jk = { t B} t A = t B t A ik t (AB)= t B t A k { } kj { } ij A= t A A = t A a 11 = a = = a = 0 a ii = a ii a ii = 0 AB= t (AB) AB= t (AB)= t B t A = BA AB = BA AB = BA AB = BA= t B t A= t (AB) AB= t (AB)
A = 1 (A+t A) + 1 (At A) P = 1 (A+t A), Q = 1 (At A) t P = 1 ( t A+ t ( t (A))) = 1 (A+t A) = P t Q = 1 ( t A t ( t (A))) = 1 (At A) = Q A = P + Q t A = t P+ t Q = P Q P = 1 (A+t A) Q = 1 (At A) A = 1 (A+t A) + 1 (At A) a 11 a 1 a 1 a 1 a a A = a 1 a a a 11 a 1 a 1 a A = 1 a a = a 1 a a = [ a 1 a a ] b 1 b b a 11 a 1 a a 1 = 1 a, a =,a = a 1 a b 1 = a 11 a 1 a 1 [ ] [ ] b = a 1 a a [ ] b = a 1 a a a 1 a a 1 0 0 1 0 0 0 1 0 E = 0, e 1 =, e = 1,e = 0 0 0 1 0 0 1
(,) (,) B = [ b 1 b b ] AB = [ Ab 1 Ab Ab ] { AB} ij = a ik b kj [ Ab 1 Ab Ab ] ij = (Ab j ) i = a ik b kj AB = [ Ab 1 Ab Ab ]