矩阵乘积的运算法则的证明

矩阵乘积的运算法则的证明矩阵乘积的运算法则1乘法结合律：若nmCA,pnCB , qpCC，则CABBCA)()(. 2乘法左分配律：若A和 B 是两个nm矩阵，且 C 是一个pn矩阵，则BCACCBA)(. 3乘法右分配律：若A是一个nm矩阵，并且 B 和 C 是两个pn矩阵，则BCACCBA)(. 4若是一个标量，并且A和 B 是两个mn矩阵，则BABA)(. 证明 1①先设 n 阶矩阵为)(ijaA,)(ijbB, )(ijcC,)(ijdAB,)(ijeBC)(ijfABC,)()(ijgBCA, 有矩阵的乘法得：njibababadnjinjijiij2,1,.2211njicbcbcbenjinjijiij2,1,.2211njicdcdcdfnjinjijiij2,1,.2211njieaeaeagnjinjijiij2,1,.2211故对任意nji2,1,有：njinjijiijcdcdcdf2211jniniicbababa11212111)(jniniicbababa22222121)(njnninninicbababa)(2211)(12121111njnjjicbcbcba)(22221212njnjjicbcbcba)(2211njnnjnjnincbcbcbanjinjijieaeaea2211=ijg故)()(BCACAB②再看mnikaA)( ,npkjbB)(,pqjtcC)(, mpijdAB)( , nqkteBC)( , mqitgBCA)()(, 有矩阵的乘法得：njibababadnjinjijiij2,1,.2211qtnkcbcbcbeptkptktkkt2,1,2,1.2211qtmicdcdcdfptiptitiit2,1,2,1.2211qtmieaeaeagntintitiit2,1,2,1.2211故对任意的,2,1mi,2,1pj,2,1nkqt2,1有：ptiptitiitcdcdcdf2211tniniicbababa11212111)(tniniicbababa22222121)(ptnpinpipicbababa)(2211)(12121111ptptticbcbcba)(22221212ptptticbcbcba)(2211ptnptntnincbcbcba6ntintitieaeaea2211=ijg故)()(BCACAB证明 2设ijA 表示矩阵A的第 i 行，第j 列上的元素，则有kjkikikijCBACBA)()(kjkikkkjikCBCA=ijijBCAC)()(故证出矩阵乘法左分配律. 证明 3同理矩阵乘法左分配律可得ijijBCAC)()(kjkikkkjikCBCAkjkikikCBA)(= ijCBA)(故证出矩阵乘法左分配律. 证明 4设mnmmnnmnijaaaaaaaaaaA212222111211)(，mnmmnnmnijbbbbbbbbbbB212222111211)(，可得BAmnmnmmmmnnnnbababababababababa221122222221211112121111，)(BA)()()()()()()()()(221122222221211112121111mnmnmmmmnnnnbababababababababaAmnmmnnaaaaaaaaa212222111211，Bmnmmnnbbbbbbbbb212222111211，BA)()()()()()()()()(221122222221211112121111mnmnmmmmnnnnbababababababababa，所以)(BA=BA.