?a1?a2?aa1(p1, p2, ???,pn)??2????????an0????an????0??? ??????????a1? 22?
?142?设A??0?34??
?043???求A100?
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)? 所以
?121?P?1??01?2??021????1?50?5?1??012?? 5?0?21???
??50?5??121??1??012? A100?1?01?2??5100??0?21?5?021??5100??????
?1??105100100??050?? ?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)? (1)求关系式??xn?1???y?n?1??x?A?n?中的矩阵A? ?yn? 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?p ??????yn?1??pq??xn??y?? 1?q???n??因此 A???p1?q??
????0? (2)设目前农村人口与城镇人口相等? 即??y???0.5?? 求
??0???xn??y?? ?n??xn?1??xn??xn?n?x0? 解 由???A?y?可知?y??A?y?? 由 y?n?1??n??n??0?
1?pqx0.5
|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? 令P?(p1, p2)???q?1?? 则 ??p1? P?1AP?diag(1? r)??? A?P?P?1? An?P?nP?1? 于是 An???q?1??10??q?1???0r??? p1p1??????n?1q?1??10??11? ?1??p1??0rn???pq?
p?q??????1?q?prnq?qrn? ??p?prnp?qrn??
p?q??1?q?prnq?qrn??0.5??xn? ????p?prnp?qrn??0.5? y??n?p?q???1?2q?(p?q)rn? ??2p?(q?p)rn??
2(p?q)??3?2?109
24? (1)设A????23?? 求?(A)?A?5A?
?? 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?
2?11?1? 于是有正交矩阵P?1???? 使得PAP?diag(1? 5)???
2?11?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???40?1?1 ?1??????11002????2??1?2?2??1 ??????2???2?2??11??? 1?1?? 1?
?212?(2)设A??122?,
?221???求?(A)?A10?6A9?5A8?
解 求得正交矩阵为
??1?31?P??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?1??20???33????2?20??2??2?0? ?2?
?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? 解
?121??x?f?(x, y, z)?242??y??
?121??????z? (2) f?x2?y2?7z2?2xy?4xz?4yz? 解
?1?1?2??x?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?
21? (1)f(x)?xT???x?
?31?21? 解 二次型的矩阵为A???31??
??