线性代数课件-第二章-矩阵及其运算——第2节.PPT

线性代数第二章矩阵及其运算１、定义mnmnmmmmnnnnbababababababababaBA221122222221211112121111一、矩阵的加法设有两个矩阵那末矩阵与的和记作，规定为nm,bB,aAijijABBA说明只有当两个矩阵是同型矩阵时，才能进行加法运算.例如12345698186309153121826334059619583112.986447411132、矩阵加法的运算规律;1ABBA.2CBACBAmnmmnnaaaaaaaaaA1122221112113.,04BABAAA,ija.负矩阵的称为矩阵A1、定义.112222111211mnmmnnaaaaaaaaaAA二、数与矩阵相乘规定为或的乘积记作与矩阵数,AAA;1AA;2AAA.3BABA2、数乘矩阵的运算规律矩阵相加与数乘矩阵合起来,统称为矩阵的线性运算.（设为矩阵，为数）,nmBA、１、定义skkjiksjisjijiijbabababac12211,,,2,1;,2,1njmi并把此乘积记作.ABC三、矩阵与矩阵相乘设是一个矩阵，是一个矩阵，那末规定矩阵与矩阵的乘积是一个矩阵，其中ijaAsmijbBnsnmijcCAB例１222263422142C221632816设415003112101A121113121430B例2?故121113121430415003112101ABC.解,43ijaA,34ijbB.33ijcC567102621710注意只有当第一个矩阵的列数等于第二个矩阵的行数时，两个矩阵才能相乘.106861985123321例如123321132231.10不存在.２、矩阵乘法的运算规律;1BCACAB,2ACABCBA;CABAACBBABAAB3（其中为数）;;4AEAAE若A是阶矩阵，则为A的次幂，即并且5nkAk个kkAAAA,AAAkmkm.mkkmAA为正整数k,m注意矩阵不满足交换律，即：,BAAB.BAABkkk例设1111A1111B则,0000AB,2222BA.BAAB故但也有例外，比如设,2002A,1111B则有,AB2222BA2222.BAAB例3计算下列乘积：213221解213221122212221323.6342423213332312322211312113212bbbaaaaaaaaabbb解332222112bababa321bbb.222322331132112233322222111bbabbabbabababa321333231232221131211321bbbaaaaaaaaabbb331221111bababa=(333223113bababa）解0010010010012A.002012222.001001kAA求设例400100100201222223AAA32323003033由此归纳出200021121kkkkkAkkkkkkk用数学归纳法证明当时，显然成立.2k假设时成立，则时，nk1nk,001001000211211nnnnnnnnnnnnAAA所以对于任意的都有k.00021121kkkkkkkkkkkA,00102111111nnnnnnnnnn定义把矩阵的行换成同序数的列得到的新矩阵，叫做的转置矩阵，记作.AAA例,854221A;825241TA,618B.618TB１、转置矩阵四、矩阵的其它运算转置矩阵的运算性质;1AATT;2TTTBABA;3TTAA.4TTTABAB例5已知,102324171,231102BA.TAB求解法1102324171231102AB,1013173140.1031314170TAB解法2TTTABAB213012131027241.10313141702、方阵的行列式定义由阶方阵的元素所构成的行列式，叫做方阵的行列式，记作或nAAA.detA8632A例8632A则.2运算性质;1AAT;2AAn;3BAAB.BAAB3、对称阵与伴随矩阵定义设为阶方阵，如果满足，即那末称为对称阵.AnTAAn,,,j,iaajiij21A.A为对称阵例如6010861612.称为反对称的则矩阵如果AAAT对称阵的元素以主对角线为对称轴对应相等.说明例6设列矩阵满足TnxxxX,,,21,1XXT.,,2,EHHHXXEHnETT且阵是对称矩证明阶单位矩阵为证明TTTXXEH2TTTXXE2,2HXXET.是对称矩阵H2HHHT22TXXETTTXXXXXXE44TTTXXXXXXE44TTXXXXE44.E例7证明任一阶矩阵都可表示成对称阵与反对称阵之和.nA证明TAAC设TTTAAC则AAT,C所以C为对称矩阵.,TAAB设TTTAAB则AAT,B所以B为反对称矩阵.22TTAAAAA,22BC命题得证.定义行列式的各个元素的代数余子式所构成的如下矩阵AijAnnnnnnAAAAAAAAAA212221212111性质.EAAAAA证明,ijaA设,ijbAA记则jninjijiijAaAaAab2211,ijA称为矩阵的伴随矩阵.A4、共轭矩阵定义当为复矩阵时，用表示的共轭复数，记，称为的共轭矩阵.ijaAijaijaijaAAA故ijAAAijA.EA同理可得nkkjkiaAAA1ijAijA.EA;2AA.3BAAB运算性质;1BABA（设为复矩阵，为复数,且运算都是可行的）:BA,五、小结矩阵运算加法数与矩阵相乘矩阵与矩阵相乘转置矩阵对称阵与伴随矩阵方阵的行列式共轭矩阵思考题问等式阶方阵为与设,nBABABABA22成立的充要条件是什么?思考题解答答,22BABBAABABA故成立的充要条件为BABABA22.BAAB