?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?