线性代数1同济五版课件2-4

第四节矩阵分块法一、矩阵的分块二、分块矩阵的运算规则三、按行分块与按列分块及有关运算四、小结思考题上页下页返回一、矩阵的分块对于行数和列数较高的矩阵，为了简化运算，经常采用分块法，使大矩阵的运算化成小矩阵的运算.具体做法是：将矩阵用若干条纵线和横线分成许多个小矩阵，每一个小矩阵称为的子块，以子块为元素的形式上的矩阵称为分块矩阵.AAA上页下页返回,321BBBbbaaA110101000001例A001aba110000b1101B2B3B即上页下页返回bbaaA110101000001,4321CCCCA1a1C002C10010a3Cbb11004C即上页下页返回,1BEOA,4321AAAAbbaaA110101000001bbaaA110101000001aaA011其中bbB111001E0000O0101aA其中1012aA1003bAbA1004上页下页返回有相同的分块法采用列数相同的行数相同与设矩阵,,,1BA那末列数相同的行数相同与其中,,ijijBA.11111111srsrssrrBABABABABA二、分块矩阵的运算规则srsrsrsrBBBBBAAAAA11111111,上页下页返回那末为数设,,21111srsrAAAAA.1111srsrAAAAA上页下页返回例654123321A,2222222222654123321A2.12108246644上页下页返回分块成矩阵为矩阵为设,,3nlBlmA,,11111111trtrststBBBBBAAAAA那末的行数的列数分别等于其中,,,,,,,2121tjjjitiiBBBAAAsrsrCCCCAB1111.,,1;,,11rjsiBACkjtkikij其中上页下页返回即是方阵且非零子块都其余子块都为零矩阵上有非零子块角线的分块矩阵只有在主对若阶矩阵为设.,,,5AnA,21sAAAAOO,411srAAA设rA11sA.11TsrTTAAA则TsA1TrA1TsA1TrA1.11TsrTTAAA则上页下页返回,21sAAAAOO.,,2,1对角矩阵为分块那末称都是方阵其中AsiAi.21sAAAA分块对角矩阵的行列式具有下述性质:上页下页返回并有则若,0,,,2,10AsiAi.21sAAAAoo,621sAAAA设oo1111上页下页返回ssBBBAAA00000000000072121.0000002211ssBABABA上页下页返回例1(第48页例14)设,1011012100100001A,0211140110210101B.AB求解分块成把BA,1011012100100001A10011001A00001121,EEO1A上页下页返回0211140110210101B11BE21B22B则2221111BBEBEAOEAB.2212111111BABBAEB上页下页返回.2212111111BABBAEBAB又21111BBA11012101112111012043,114202141121221BA,1333上页下页返回于是2212111111BABBAEBAB.1311334210410101上页下页返回,100100000001bbaaA设bbaaB100000001000.,ABABA求例2(补充例题)上页下页返回解分块将BA,bbaaA100100000001,0021AAbbaaB100000001000,0021BB其中,011aaA;112bbA,101aaB;102bbB其中上页下页返回21210000BBAABA,002211BABAaaaaBA100111,2112aabbbbBA101122,2212bb上页下页返回.2200120000210012bbaa21210000BBAABA221100BABA上页下页返回212121000000AABBAAABA,00222111ABAABA,123223111aaaaaaABA,231223223222bbbbbbABA上页下页返回212121000000AABBAAABA22211100ABAABA.23001220000001232233223bbbbbbaaaaaa上页下页返回例3(第49页例15)设,120130005A.1A求解120130005A,21AOOA,51A;5111A,12132A上页下页返回;321112A12111AOOAA;5111A.3201100051上页下页返回三、按行分块与按列分块及有关运算m×n矩阵A有m行，称为矩阵A的m个行向量。若第i行记作则矩阵A便可记作,,,,21iniiTiaaa.21TmTTAm×n矩阵A有n列，称为矩阵A的n个列向量。若第j列记作则矩阵A便可记作,21njjjjaaaa.,,,21naaaA上页下页返回对矩阵A=(aij)m×s与矩阵B=(bij)s×n的乘积矩阵AB=C(cij)m×n，若把矩阵A按行分成m块，把矩阵B按列分成n块，便有.,,,2122212121112121nmijnTmTmTmnTTTnTTTnTmTTcbbbbbbbbbbbbAB其中由此可进一步领会矩阵相乘的定义。,,,,12121skkjiksjjjisiijTiijbabbbaaabc上页下页返回以对角矩阵Λm左乘矩阵A=(aij)m×n时,把A按行分块，有,22112121TmmTTTmTTmnmmA可见以对角矩阵Λm左乘Am×n的结果是A的每一行乘以Λm中与该行对应的对角元。以对角矩阵Λn右乘矩阵A=(aij)m×n时,把A按列分块，有,,,,,,,22112121nnnnmaaaaaaA可见以对角矩阵Λ右乘A的结果是A的每一列乘以Λ中与该列对应的对角元。上页下页返回例4(第51页例16)设ATA=O，证明A＝O。证明设A＝（aij）m×n，把A用列向量表示为：,,,,2122212121112121nTnTnTnnTTTnTTTnTnTTTaaaaaaaaaaaaaaaaaaaaaaaaAA由已知ATA=O，故,,,,21naaaA),,...,2,1,(,0njiaajTi特殊地，有),,...,2,1(,0,,,122121niaaaaaaaaamkkimiiimiiiiTi上页下页返回从而即A=O。),...,2,1(,021niaaamiiimnmnmmnnnnbxaxaxabxaxaxabxaxaxa22112222212111212111对于线性方程组,,,,2121bABbbbbxxxxaAmnnmij记四、线性方程组的矩阵表示（1）上页下页返回其中A称为系数矩阵，x称为未知数向量，b称为常数项向量，B称为增广矩阵。利用矩阵乘法，此方程组可记作（2）bAx方程组（2）以向量x为未知元，它的解称为线性方程组（1）的解向量。,,221121mTmTTTmTTbxabxabxabxaaa或若把系数矩阵A按行分成m块，则线性方程组（2）可记作（3）若把系数矩阵A按列分成n块，则线性方程组（2）可记作上页下页返回baxaxaxbxxxaaannnn22112121,,,,即（4）(2)、(3)、(4)是线性方程组（1）的各种变形。今后他们将与（1）混同使用而不加区别。五、克拉默法则的证明对于n个变量、n个方程的线性方程组nnnnnnnnnnbxaxaxabxaxaxabxaxaxa22112222212111212111（5）上页下页返回如果它的系数行列式D≠0，则它有唯一解证把方程组（5）写成向量方程bAx这里A=(aij)n×n为n阶方阵，因，故A-1存在。.),,2,1(,12211njAbAbAbDDDxnjnjjjj0DA令,1bbAAAx有,1bAx表明是方程组（5）的解向量。bAx1,,111bAxbAAxA即由，有bAx根据逆矩阵的唯一性，知式方程组（5）的唯一解向量。bAx1