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1、线性代数 第二章 矩阵 张祥朝 复旦大学光科学与工程系 2013-3-19 对角形矩阵 数量矩阵 单位矩阵 三角形矩阵 对称矩阵与反对称矩阵 正交矩阵 对角矩阵(diagonal) = = = = a a a A O 22 11 )(0jiaij = = = = 注 nn a ( ( ( () ) ) ) nn aaadiagAL，记为记为记为记为： 2211 = = = = 不是对角阵不是对角阵不是对角阵不是对角阵 0300 0020 0001 性质（1）A,B为n阶对角阵，k为常数 .,BAABBAABBAkA= = = =+ + + +仍为对角阵仍为对角阵仍为对角阵仍为对角阵，且且且且
2、因 = = = = nnnn b b b a a a AB OO 22 11 22 11 AAT= = = =)2( aaaAL|)3(= = = = = nnnnb a ba ba O 2222 1111 BA= = = = AA)2( nn aaaAL 2211 |)3(= )( 11 = i adiagA（4）若A可逆，则 （5）若设, 则 T nn C),(),( 2121 LL= = nn a a a AC M 22 11 nn aaaCAL 2211 = （二）数量矩阵 = = = = a a a A O 对角元素相等；对角阵的 特别情形 性质： ln nlnn l l nn bb
3、b bbb bbb a a a AB = = = = L LL L L O 21 22221 11211 ln nlnn l l ababab ababab ababab = = = = L LL L L 21 22221 11211 aB= = = = 单位阵 IAA= = = = = = = ? 1| IAIA= = = = = = = ?2 = = = = 21 11 A反例反例反例反例： = = = = 2 1 2 1 2 1 2 1 A反例反例反例反例： )(? 2?)( 22 222 IAIAIA IAIAIA + + + + + + + + + + + + + + 22 等式成立
4、 推广 例设 n AA，计算计算计算计算 = = = = 111 011 001 解法二 + + + + = = = = = = = = 011 001 000 100 010 001 111 011 001 AQ BI + + + += = = = OB = = = = 3 nn BIA)( + + + += = = =InBB nn B n + + + + + + + + + + + + + += = = = 2 2 )1( L = = = = 001 000 000 2 B InBB nn + + + + + + + = = = = 2 2 )1( + + + + = = = = 1 2
5、 )1( 01 001 n nn nA n 例 . , 222 CBA ICABCABnCBA + + + + + + + = = = = = = = = = = 求求求求 阶方阵阶方阵阶方阵阶方阵，且且且且为为为为、设设设设 解 AIAA = = = = 2 ABCA)(= = = =)(CAAB= = = =II= = = =I= = = = BIBB = = = = 2 BCAB)(= = = =)(ABBC= = = = II= = = =I= = = = 单位矩阵的妙用 练习： CABC)(= = = = CICC = = = = 2 )(BCCA= = = = II= = = =I
6、= = = = 所以 ICBA3 222 = = = =+ + + + + + + 2 C求求求求 例设 A 是n阶方阵， 82)( 2 + + + += = = =xxxf 82)( 2 + + + += = = =AAAf则则则则I 称为矩阵A的一个多项式。 矩阵多项式 )82)( 2 + + + += = = =AAAf * 称为矩阵A的一个多项式。 )4)(2(82 2 + + + + = = = = + + + +xxxx因因因因 )4)(2(82 2 IAIAIAA+ + + + = = = = + + + +所以所以所以所以 矩阵多项式的分解： 分解*式 三角矩阵 nn n n
7、a aa aaa L MMMM L L 00 0 222 11211 nnnn aaa aa a L OMM 21 2221 11 0 00 000 性质: A、B是同阶、同型的三角形矩阵 kA, A+B, AB 仍是同阶同型三角形矩阵。 三角阵的逆矩阵为同型三角阵 对称矩阵 是对称矩阵是对称矩阵是对称矩阵是对称矩阵称称称称)( ij aA = = = =，若若若若 jiij aa = = = = ，若若若若 jiij aa = = = = 是反对称矩阵是反对称矩阵是反对称矩阵是反对称矩阵称称称称)( ij aA = = = = )0(= = = = ii a AAT= = = =性质：（1）
8、A 是对称阵 （ ）是 阶对称阵 是对称阵是对称阵是对称阵是对称阵BA+ + + + A是反对称阵 AAT = = = = （2）A,B是n阶对称阵 是对称阵是对称阵是对称阵是对称阵BA+ + + + AB 是对称阵 对称矩阵的方幂、逆矩阵仍为对称阵 A,B是n阶对称阵 A,B是n阶对称阵，则 BAABAB= = = =对称对称对称对称 证 BBAA TTTTT ABAB)( BA= = = =AB= = = = ”“ ,BBAA TT = = = = = = =Q TTT ABAB= = = =)( BA= = = =AB= = = = 是对称阵是对称阵是对称阵是对称阵。故故故故AB ”“
9、T ABAB)(= = = =是对称阵是对称阵是对称阵是对称阵，ABQ TT ABBA= = = = 设A与B为两个n 阶矩阵，A为反对称矩阵， B为对称矩阵，则AB-BA为对称矩阵。 证 )( T BAAB TTTT TT BAAB)()( = = = = BBAA TT = = = = = = = =,由已知条件得由已知条件得由已知条件得由已知条件得： 为对称矩阵为对称矩阵为对称矩阵为对称矩阵。即即即即BAAB ，AB-BA= = = = BAAB)()( = = = = TTTT BAAB = = = = 正交矩阵(orthogonal matrix) A为正交矩阵：ATA=AAT=E 性质： (1) A-1=AT (2) 按行或列分块： (3) de
