2.4-分块矩阵

1第2.4节分块矩阵2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一、分块矩阵的定义3一、分块矩阵的定义二、分块矩阵的运算规则主要内容:三、思考与练习4,321BBBbbaaA110101000001例：A001aba110000b1101B2B3B即5bbaaA110101000001,4321CCCCA1a1C002C10010a3Cbb11004C即6,BEOA,4321AAAAbbaaA110101000001bbaaA110101000001aaA01其中bbB111001E0000O0101aA其中1012aA1003bAbA10047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srsrsrsrBBBBBAAAAA11111111,二、分块矩阵的运算规则8那末为数设,,21111srsrAAAAA.1111srsrAAAAA9例654123321A,2222222222654123321A2.1210824664410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11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12,21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:13并有则若,0,,,2,10AsiAi.21sAAAAoo,621sAAAA设oo111114ssBBBAAA00000000000072121.0000002211ssBABABA15例1：设,1011012100100001A,0211140110210101B.AB求解分块成把BA,1011012100100001A10011001A00001121,EEO1A160211140110210101B11BE21B22B则2221111BBEBEAOEAB.2212111111BABBAEB17.2212111111BABBAEBAB又21111BBA11012101112111012043,114202141121221BA,133318于是2212111111BABBAEBAB.131133421041010119例2：设,120130005A.1A求解：120130005A,21AOOA,51A;5111A,12132A20;321112A12111AOOAA;5111A.320110005121,0*AABEPT,bBBAQT试证:BABbAPQT12例3:是常数，矩阵，是阶非奇异矩阵，是设bnBnA1的伴随矩阵，记是AA22证由bBBAAABEPQTT*0bABABBAAABBATTT**bABAABBAT10,01BABbABAT知BABbAPQT1223在矩阵理论的研究中,矩阵的分块是一种最基本,最重要的计算技巧与方法.(1)加法:采用相同的分块法同型矩阵,(2)数乘:的每个子块乘需乘矩阵数AkAk,(3)乘法:,ABAB若与相乘需的列划分与的行划分相一致分块矩阵之间与一般矩阵之间的运算性质类似。小结：(4)转置:srAAA11rA11sATsA1TrA1TsrTTAAA1124(5)分块对角阵的行列式与逆阵sAAAA21OO.21sAAAA.,,,,,2,1112111siAAAdiagAsiAA且可逆可逆11121sAAAA－－－OO25,,0都是可逆方阵和其中设CBCDBA1,.AA可逆并求证明：三、思考与练习证：,,可逆由CB,0CBA有.可逆得A,1YWZXA设.000EEYWZXCDB则.,,,ECYOCWODYBZEDWBX.,,,1111OWDCBZCYBX11111.BBDCAOC