同济大学线性代数课件第二章

1第二章矩阵及其运算2§1矩阵979634226442224321432143214321xxxxxxxxxxxxxxxx97963422644121121112线性方程组与矩阵的对应关系3)2121(1njmianmij,,,;,,,个数由定义列的数表，行排成的nm.列矩阵行称为nm.mn简称矩阵111212122212nnmmmnaaaaaaaaa4mnmmnnaaaaaaaaaA212222111211记作简记为ijmnAanmA或其中数ija称为mnA的第i行第j列的元素,nmA或的(i,j)元素。5420134081zyx同型矩阵：两个矩阵的行数相等、列数也相等。.),,,,()(,)(BABAnjibabBaABAijijijij相等，记作与则称矩阵若是同型矩阵，与设矩阵21矩阵相等：823zyx,,6一些特殊的矩阵零矩阵(ZeroMatrix):注意：.0000000000不同阶数的零矩阵是不相等的.元素全为零的矩阵称为零矩阵，零矩阵记作或.nmmnOO7行矩阵(RowMatrix):列矩阵(ColumnMatrix):只有一行的矩阵,,,,21naaaA称为行矩阵(或行向量).,naaaA21只有一列的矩阵称为列矩阵(或列向量)8方阵(SquareMatrix):302234163是3阶方阵.行数与列数都等于n的矩阵，称为n阶方阵(或n阶矩阵),记作An9对角阵(DiagonalMatrix):主对角线以外的元素都为零的方阵。nn2121),,(diag10数量矩阵(ScalarMatrix):nnnkkkEk主对角元素全为非零常数k，其余元素全为零的方阵。11单位矩阵(IdentityMatrix):)(jinnnE111主对角元素全为1，其余元素都为零的方阵。记作:EEn或jijiji0112例3：11111221221122221122nnnnmmmmnnyaxaxaxyaxaxaxyaxaxax从变量nxxx,,21到变量myyy,,21的线性变换.其中ija为常数.称为系数矩阵nmijaA)(13线性变换与矩阵之间的对应关系.nnxyxyxy,,2211100010001恒等变换单位阵nnnxyxyxy222111n2114§2矩阵的基本运算一、矩阵的加法mnmnmmmmnnnnbababababababababaBA221122222221211112121111设有两个矩阵那末矩阵A与B的和记作A+B，规定为nmijijAaBb(),(),定义215注意：只有当两个矩阵是同型矩阵时，才能进行加法运算.12345698186309153121826334059619583112.9864474111316负矩阵：)(BABAmnmmnnaaaaaaaaaA112222111211ija),(jiaA设称为矩阵A的负矩阵。17矩阵加法满足的运算规律:.1ABBA交换律：.2CBACBA结合律：.4OAA3AOA18二、数与矩阵相乘.112222111211mnmmnnaaaaaaaaaAA规定为或的乘积记作与矩阵数,AAA定义3191101013213131303)1(3031333231333030396320;1AA;2AAA.3BABA数乘矩阵满足的运算规律：矩阵相加与数乘矩阵运算合起来,又称为矩阵的线性运算.设A,B为m×n矩阵，,为数AAA11421定义4skjkkijssijijijibabababac12211),,,;,,(njmi2121并把此乘积记作C=AB三、矩阵与矩阵相乘设是一个m×s矩阵，是ijAa()ijBb()一个s×n矩阵，那末规定矩阵A与矩阵BijCc()的乘积是一个m×n矩阵，其中ss221331654321635241321331124563334568101212151823例：2222634221422216328164331200311210142102111241321013212432133322211111111111111111111132132132132125nnnnnnbbbaaa2121nnnnbababa2211261.矩阵乘法不满足交换律.BAAB注意：11111111AB1111A1111B000011111111BA2222设A左乘BB右乘A272.矩阵乘法不满足消去律OAACAB,1111A1111B设0000C11111111AB000000001111AC0000CB但注意：28nmnmmmnnnnxaxaxayxaxaxayxaxaxay22112222121212121111mnmmnnaaaaaaaaaA11222211121112(,,,)myyyy12(,,,)nxxxxyAx29矩阵乘法满足的运算规律：;:1BCACAB结合律,:2ACABCBA分配律;CABAACBBABAAB3;4AEAAE30若A是n阶方阵，则为A的次幂，即kAk个kkAAAA,klklAAA.klklAA方阵的幂：并且,时但当BAAB.BAABkkk31方阵的多项式：0111)(axaxaxaxkkkkEaAaAaAaAkkkk0111)(1011A52)(3xxx10015101121011)(3A401432例.设332313322212312111bababababababababaA321321332313322212312111bbbaaabababababababababa求nAnn33321321321321321321bbbaaabbbaaabbbaaa3213211332211)(bbbaaabababan34四.矩阵的转置定义:把矩阵A的行换成同序数的列得到的新矩阵，叫做A的转置矩阵，记作.A例:,854321A;835241ΤA35转置矩阵满足的运算规律：;)()1(TTAA;)()2(TTTBABA;)()3(TTAA.)()4(TTTABAB36例5：已知,102324171,231102BAT)(AB求37解1：102324171231102AB,1013173140.1031314170TAB38解2：TTTABAB213012131027241.103131417039对称阵的元素以主对角线为对称轴。对称阵:设A为n阶方阵，如果满足，即那末A称为对称阵.njiaaijji,,2,1,AAT304021411A40反对称阵:设A为n阶方阵，若满足，即则称A为反对称阵.njiaaijji,,2,1,AAT024201410A显然,反对称阵的主对角元都是零。41例。与反对反对阶矩阵，证明是设对称矩阵之和称矩阵可表示为对称矩阵是称矩阵是AAAAAnA2,,1:TT注：对称矩阵的乘积不一定是对称矩阵31112111110000101031111112142五、方阵的行列式定义：由n阶方阵A的元素所构成的行列式，叫做方阵A的行列式，记作|A|或detA110101321:A例110101321A则243运算规律：;1TAA;2AAnBAAB3.ABBA注：虽然,ABBA但44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称为矩阵A的伴随矩阵.T)(jiA4511010132112111332313322212312111AAAAAAAAA114246jiAAA故jiAEAninjijijAaAaAaAA2211同理EA性质：EAAAAA,jiaA设,jibAA记jninjijijiAaAaAab2211则,jiA47§3逆矩阵定义：设A是n阶矩阵，若存在n阶矩阵B使AB=BA=E则称A是可逆的，并称B是A的逆矩阵，111121,1111BA48CCEABCBCAEBBECAACEBAABACB)()(从而，的逆矩阵，则都是、设若A是可逆矩阵，则A的逆矩阵是唯一的。记A的逆矩阵为1A49定理1:证明：n阶方阵A可逆充要条件是|A|0,且当A可逆时,AAA||11A可逆,存在B