矩陣乘積的運(yùn)算法則的證明

1、矩陣乘積的運(yùn)算法則的證明矩陣乘積的運(yùn)算法則乘法結(jié)合律：若，則.?1nmCA??pnCB??qpCC??CABBCA)()(?乘法左分配律：若和是兩個(gè)矩陣，且是一個(gè)矩陣，則?2ABnm?Cpn?.BCACCBA???)(乘法右分配律：若是一個(gè)矩陣，并且和是兩個(gè)矩陣，則?3Anm?BCpn?.BCACCBA???)(若是一個(gè)標(biāo)量，并且和是兩個(gè)矩陣，則.?4?ABmn?BABA??????)(證明證明?1①先設(shè)階矩陣為n)(ijaA?)(ij

2、bB?)(ijcC?)(ijdAB?)(ijeBC?有矩陣的乘法得：)(ijfABC?)()(ijgBCA?njibababadnjinjijiij??21.2211?????njicbcbcbenjinjijiij??21.2211?????njicdcdcdfnjinjijiij??21.2211?????njieaeaeagnjinjijiij??21.2211?????故對(duì)任意有：nji?21?njinjijiijcdcdcdf

3、?????2211?????jniniicbababa11212111)(?????jniniicbababa22222121)(?njnninninicbababa)(2211???????????)(12121111njnjjicbcbcba?故)()(BCACAB?證明證明?2設(shè)表示矩陣的第行，第列上的元素，則有ijAAij??kjkikikijCBACBA????)()(kjkikkkjikCBCA????=ijijBCAC)(

4、)(?故證出矩陣乘法左分配律.證明證明?3同理矩陣乘法左分配律可得ijijBCAC)()(?kjkikkkjikCBCA????kjkikikCBA???)(=??ijCBA)(?故證出矩陣乘法左分配律.證明證明?4設(shè)，，??????????????mnmmnnmnijaaaaaaaaaaA??????212222111211)(??????????????mnmmnnmnijbbbbbbbbbbB??????212222111211)