线性代数第五章习题

?5??1?2?4??17? 设矩阵A???2x?2?与???4?相似? 求x? y? 并求一

??4?21???y????个正交阵P? 使P?1AP???

18? 设3阶方阵A的特征值为?1?2? ?2??2? ?3?1? 对应的特征向量依次为p1?(0? 1? 1)T? p2?(1? 1? 1)T? p3?(1? 1? 0)T? 求A.

19? 设3阶对称阵A的特征值为?1?1? ?2??1? ?3?0? 对应?1、?2的特征向量依次为p1?(1? 2? 2)T? p2?(2? 1? ?2)T? 求A?

20? 设3阶对称矩阵A的特征值?1?6? ?2?3? ?3?3? 与特征值?1?6对应的特征向量为p1?(1? 1? 1)T? 求A. 21? 设a?(a1? a2? ???? an)T ? a1?0? A?aaT? (1)证明??0是A的n?1重特征值?

(2)求A的非零特征值及n个线性无关的特征向量?

?142?22? 设A??0?34?? 求A100?

?043???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?x0??0.5? (2)设目前农村人口与城镇人口相等? 即??y???0.5?? 求

?0????xn?? ?y??n?3?2? 求?(A)?A10?5A9? 24? (1)设A????23?????212? (2)设A??122?, 求?(A)?A10?6A9?5A8?

?221???25? 用矩阵记号表示下列二次型: (1) f?x2?4xy?4y2?2xz?z2?4yz? (2) f?x2?y2?7z2?2xy?4xz?4yz?

(3) f?x12?x22?x32?x42?2x1x2?4x1x3?2x1x4?6x2x3?4x2x4? 26? 写出下列二次型的矩阵?

2 (1)f(x)?xT??3?1?x? 1???123? (2)f(x)?xT?456?x?

?789???27? 求一个正交变换将下列二次型化成标准形: (1) f?2x12?3x22?3x33?4x2x3?

(2) f?x12?x22?x32?x42?2x1x2?2x1x4?2x2x3?2x3x4?

28? 求一个正交变换把二次曲面的方程

3x2?5y2?5z2?4xy?4xz?10yz?1 化成标准方程?

29? 明? 二次型f?xTAx在||x||?1时的最大值为矩阵A的最大特征值.

30? 用配方法化下列二次形成规范形? 并写出所用变换的矩阵 (1) f(x1? x2? x3)?x12?3x22?5x32?2x1x2?4x1x3? (2) f(x1? x2? x3)?x12?2x32?2x1x3?2x2x3? (3) f(x1? x2? x3)?2x12?x22?4x32?2x1x2?2x2x3?

31? 设

f?x12?x22?5x32?2ax1x2?2x1x3?4x2x3

为正定二次型? 求a?

32? 判别下列二次型的正定性? (1) f??2x12?6x22?4x32?2x1x2?2x1x3?

(2) f?x12?3x22?9x32?19x42?2x1x2?4x1x3?2x1x4?6x2x4?12x3x4? 33? 证明对称阵A为正定的充分必要条件是? 存在可逆矩阵U? 使A?U TU? 即A与单位阵E合同?