?142? 22? 设A??0?34?? 求A100?
?043??? 解 由
1??42 |A??E|?0?3??4??(??1)(??5)(??5)?
043??得A的特征值为?1?1? ?2?5? ?3??5?
对于?1?1? 解方程(A?E)x?0? 得特征向量p1?(1? 0? 0)T? 对于?1?5? 解方程(A?5E)x?0? 得特征向量p2?(2? 1? 2)T? 对于?1??5? 解方程(A?5E)x?0? 得特征向量p3?(1? ?2? 1)T? 令P?(p1? p2? p3)? 则
P?1AP?diag(1? 5? ?5)??? A?P?P?1? A100?P?100P?1? 因为
?100?diag(1? 5100? 5100)? ?121??50?5?1 P?1??01?2???012??
?021?5?0?21??????1所以
??50?5??121??11??012? A100??01?2??51005?021??100??0?21?5???????105100?1? ??051000??
?005100??? 23? 在某国? 每年有比例为p的农村居民移居城镇? 有比例为q的城镇居民移居农村? 假设该国总人口数不变? 且上述人口迁移的规律也不变? 把n年后农村人口和城镇人口占总人口的比例依次记为xn和yn(xn?yn?1)?
xn?1??xn? (1)求关系式??y??A?y?中的矩阵A?
?n?1??n? 解 由题意知
xn?1?xn?qyn?pxn?(1?p)xn?qyn? yn?1?yn?pxn?qyn? pxn?(1?q)yn? 可用矩阵表示为
xn?1??1?pq??xn? ??y???p1?q??y??
??n??n?1??1?pq?因此 A???p1?q??
??x??0.5? (2)设目前农村人口与城镇人口相等? 即??y0???0.5?? 求
?0????xn?? ?y??n?xn?1??xn??xn??An?x0?? 由 ?A 解 由?可知?y??y??y??y??n?1??n??n??0?|A??E|?1?p??q?(??1)(??1?p?q)?
p1?q??得A的特征值为?1?1? ?2?r? 其中r?1?p?q?
对于?1?1? 解方程(A?E)x?0? 得特征向量p1?(q? p)T? 对于?1?r? 解方程(A?rE)x?0? 得特征向量p2?(?1? 1)T? q?1? 令P?(p1, p2)???p1?? 则
?? P?1AP?diag(1? r)??? A?P?P?1? An?P?nP?1?
q?1??10?于是 An???p1??0r?????n?q?1?
?p1????1q?1??10??11? ?1??p1??0rn???pq?
p?q??????q?prnq?qrn?1? ??p?prnp?qrn?? p?q??xn?q?prnq?qrn??0.5?1?? ????p?prnp?qrn??0.5? y?n?p?q????2q?(p?q)rn?1? ???? 2(p?q)?2p?(q?p)rn?3?2? 求?(A)?A10?5A9? 24? (1)设A????23???? 解 由
|A??E|?3???2?(??1)(??5)?
?23??得A的特征值为?1?1? ?2?5?
对于?1?1? 解方程(A?E)x?0? 得单位特征向量1(1, 1)T?
2 对于?1?5? 解方程(A?5E)x?0? 得单位特征向量1(?1, 1)T?
21?1? 使得P?1AP?diag(1? 5)??? 于是有正交矩阵P?1??11??2??从而A?P?P?1? Ak?P?kP?1? 因此 ?(A)?P?(?)P?1?P(?10?5?9)P?1 ?P[diag(1? 510)?5diag(1? 59)]P?1 ?Pdiag(?4? 0)P?1 1?1???4 ?1????2?11??00?1?11?
??0??2??11??2?2??2?11?? ????2?2???11??????212? (2)设A??122?, 求?(A)?A10?6A9?5A8?
?221??? 解 求得正交矩阵为
??1?3P?1??13?6?20?2?? 2??2??使得P?1AP?diag(?1? 1? 5)??? A?P?P?1? 于是 ?(A)?P?(?)P?1?P(?10?6?9?5?8)P?1 ?P[?8(??E)(??5E)]P?1
?Pdiag(1? 1? 58)diag(?2? 0? 4)diag(?6? ?4? 0)P?1 ?Pdiag(12? 0? 0)P?1
??1?31 ???13?6?20?2??12???1?12??2??0???330? ???222?2?0??????11?2? ?2?11?2??
??2?24??? 25? 用矩阵记号表示下列二次型: (1) f?x2?4xy?4y2?2xz?z2?4yz? ?121??x? 解 f?(x, y, z)?242??y??
?121??z????? (2) f?x2?y2?7z2?2xy?4xz?4yz? ?1?1?2??x? 解 f?(x, y, z)??11?2??y??
??2?2?7??z????? (3) f?x12?x22?x32?x42?2x1x2?4x1x3?2x1x4?6x2x3?4x2x4? ?1??1 解 f?(x1, x2, x3, x4)?2??1??113?22310?1??x1??2??x2?? 0??x3???1???x4? 26? 写出下列二次型的矩阵? 2 (1)f(x)?xT??3?1?x? 1??2 解 二次型的矩阵为A???3?1?? 1???123? (2)f(x)?xT?456?x?
?789????123? 解 二次型的矩阵为A??456??
?789???