西安毛子科技大学XIDIANUNIVERSITY矩阵乘法的运算规律2.(1)(结合律)(AB)C = A(BC)(2)A(B+C) = AB+ AC(分配律)(B+C)A = BA+CA(3)AsxE, = E,Asxn = AsxXI(4)0A=0A0 = 0,a(5)h
2．矩阵乘法的运算规律 (1) ( ) ( ) AB C A BC = (2) ( ) A B C AB AC + = + (3) A E E A A s n n s s n s n = = ( ) B C A BA CA + = + (5) 1 1 1 1 n n n n a b a b a b a b = (结合律) （分配律) (4) 0 0, 0 0 A A = =
西安毛子科技大学三XIDIAN UNIVERSITY证: 1) 设 A=(a,)sn， B=(bik)mm C=(cu)mr今V=AB=(vik)sm’ W = BC=(Wi)nr其中 Vi=之apbiu， Wu=bucu.k=1i-1W-(abunVC的第i行第1列元素为CVikCuk=1ik=1-ZZapbucuk-1 j-1-Za,(ZbiZaw,CuAW的第i行第1列元素为j-1j=1k=1abucu=bjkCklaij结合律得证。k-1 j-1j-1 k=1
证：1）设 ( ) , ( ) , ( ) A a B b C c = = = ij sn jk nm kl mr 令 ( ) , ( ) , V AB v W BC w = = = = ik sm jl nr 其中 1 1 , . n m ik ij jk jl jk kl j k v a b w b c = = = = VC 的第 i 行第 l 列元素为 1 1 ( ) m n ij jk kl k j a b c = = = 1 m ik kl k v c = 1 1 m n ij jk kl k j a b c = = = AW 的第 i 行第 l 列元素为 1 n ij jl j a w = 1 1 ( ) n m ij jk kl j k a b c = = = 1 1 n m ij jk kl j k a b c = = = 1 1 . m n ij jk kl k j a b c = = = 结合律得证