[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级线性空间[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex]上的线性变换．１）若[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]在[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex]的某组基下矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是某多项式[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]的伴侣阵，则[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的最小多项式是[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]．２）设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的最高次的不变因子是[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]，则[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的最小多项式是[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]．

2022-07-24

[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级线性空间[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex]上的线性变换．１）若[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]在[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex]的某组基下矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是某多项式[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]的伴侣阵，则[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的最小多项式是[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]．２）设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的最高次的不变因子是[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]，则[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的最小多项式是[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]．

答案：

证:１） 由题意，设[tex=14.643x1.5]6Di+DmdQwmGC+YRb94SC/fuXKAypgf1dvfjY9IoT1Id9T1cNcEo4JIqWYyjOtgmUq0ciGuxJyvm3uMtmVf2EfeDnVDGVG0gXWCB5oZUvwSY=[/tex]则，[tex=12.429x6.5]hB8sGfF8hpZRTKdvt1J/eHhfyS7QNgsPb4Linxq0W+3sy2rcYBY8YEvDxVi2Nci4+jzEVbHGJ3P2RbbyfI+3ZVMWBiAeiBIH+Nvi16X8hiQC/wbaej7OMDWPvFq2qQO+6yXrVstfcnL5TfiwMB/9SlGO7IB7R+/a7uZvmP1C/OvyR0oW2nli1WJMW5AR6PZjGUCCiJ9rAQ2922B7Ut1jUQRuG7KSdPLwCT0dTI/baQsEaj5RNRUYOSUXTBeFo0uD[/tex]由本章习题３知，[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的不变因子为     [tex=4.643x1.286]X6v+/5SaPrWKDRNDt1El8VUiuR6i0Vouv0bsvMzb80s=[/tex][tex=6.714x1.357]p2oVGfUQ7Z0fhlRBvv7fcgxU3Zfo8+GLyC+xVR0sHbs=[/tex]再设，[tex=15.643x1.357]F4S7+0+ks22+odBGU7vZR/HBhhAZeebw4f1/mhlvkte0VYi3Ke9M51A2gxaAjkcq6uUCXBx5F7beaEgjGB3D8pZwfZTLg9FTZVXI3lP5YBkgAFFIsXB96TpN1NizreYIuvu5KfMoStXbk/RKlxm/I9adkQ4caayEoXqObFfN3UhaqsuaCBYIeNHi9U8OOJKa[/tex]其中，[tex=14.357x1.357]G+Ybd2CU4QmoQJBJqfY6xKiwzjSfdCebrGDuRpKh4TGIAjS07Nhh7vQb2NCVnPACuJUVLl2PjIh9FVpEmzq/hEPHOMPFIpTTlfjbdKJXfCw=[/tex]则[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的初等因子组是[tex=14.929x1.357]e4dkp9ttpHVj3C/YhDjS+hXZ0IoLCxFOl5ZIK8GOUF/fXF+Ft0Et/BdSzWeCkk/y9MlI6cZNd2dgY9+EsXsqMAjfBuGRdlVG7QPcIrTExJfMLsT8a0NuGaR3WZUtn0A78FKNixnQyuRdqnXzq0OzAdpvy52saGY6mLGwKkpelnmAMuX6KCmZKQEz3fsB+YAM[/tex][tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的若尔当标准形为[tex=11.929x5.214]icYG/GDpJWkyHUlVRZlJy+xixtwhIYrGRUdVbEoqJdSx0NOt+dwpX3NCY2RFjIxanlakSYasPNYvp5L64LgqC2CfunAyky3kcWvwtKWfjVlS8TxgVaXBf0wS8Ss5q/qHikq6P/Fwmb27p5+oMzKu7wpUSB5b938p0IgH+8xZQFE=[/tex]其中[tex=18.286x5.214]MV9Ivi6nZFLThvHSS1fCQgSbFP4kibd4Bifz1ATCAN2cYolrURX6johFg0Z7PVjbvU/qie10Cm6V0Bd1TpZ1sDyl4XCwno4d0670BRkVPxkE59UwIXfGHSJ46qeY3T6rPB8t9ufavHKGGBLST0/cUde1EP2O6Kp8g+PgHHFSLnzmFiW5irx1eKw+PNWx1Dby945VkIbyMhJ0WfnCvk4QCScUEpBQ1TYO6lSUhHRQew8=[/tex]设[tex=6.857x1.357]oVFFBA7cF+8BHqdZoNYkc/tYc6OLtma+PDTGfQwtzVQ=[/tex]，则有[tex=10.643x1.429]IYOHuBckoSFqQl5AdI+mxDemdLUb3bPJmtMw3bYfQ6om740bgwrg43p5llPy4+d0LXwCye2N38P4HDH5hwfxjA==[/tex]，所以[tex=20.357x1.5]A0JAuf967vco+fZcYsTZLnS2VTmizcJjZAtLasSi7+4yreNctfSV0/4nqEr4jb5OTKIlWa51bTbguOoWmYqaTw==[/tex]可见相似矩阵有相同的最小多项式，由第七章９引理３和引理 知，[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]（或[tex=0.571x1.0]qmbwF4Pp2sLBvOFTeKQ/mA==[/tex]） 的最小多项式为[tex=15.643x1.357]e4dkp9ttpHVj3C/YhDjS+hXZ0IoLCxFOl5ZIK8GOUF+LcXglix9shWIyIKoJO4+fQzhhRjpVJQGhlx8wA4Z4aJAOxqXlaEs5jYAZVA7DAj9XTo8Z2ExN4syMMd3DOWLzg/KXNBca/Vj9LQ5THp7/0OKDaSeLSJ3mavT3+9v6xJytW2JD0oxoanCxmsUitpxQ[/tex]从而[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的最小多项式也为[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]．２） 设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的最高次的不变因子是[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]，由于[tex=9.214x1.357]mkF748fAcFtDAHarSpAxwLhqTe1Yieo28q03WAjR1+RZ21Sp7k6/1JY5oAZBZbqyeapliklYhUE041eux4Tzhw==[/tex]所以，[tex=5.0x1.357]cXErbJChxbwNiACS3P1qOBu7VjhxTS02xioYTGXyP4o=[/tex]又根据由非常数不变因子可得到[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]在[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex]的某组基下的有理标准形，由 １）的结果和第七章９引理３知，[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的最小多项式是[tex=2.357x1.357]jJv0yNIpJ2+sbmiMC3wFBA==[/tex]，故[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的最高次 不变因子[tex=1.929x1.357]EJ5ekqmr2bWoAT+xH4aA4Q==[/tex]就是[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的最小多项式．

举一反三

内容

0
设3阶矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的特征值为-2, -1, 3,矩阵[tex=6.786x1.357]5sQBSCH1+oEoQda8DcapHw==[/tex],求矩阵[tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]的行列式[tex=1.357x1.357]JRr5OoiiAPF9KB2ukKJtuw==[/tex]
1
设有向量组[tex=8.071x1.214]Mdl5SvJLPWwKArgK4Ta6j3l7EaXa+zhJXo0rPe0F/fLHhrYOFnnWnQKmBtyXiEqBbljE4xNvGj0KKJpF/wCa9Jqzol6QqJ+jQIfh4xKmXNjLM2WgDkUXj9CtB5g71A74[/tex]，证明(1)[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的任何部分组线性相关，则整体组线性相关；(2)向量组[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]线性相关，则[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的任何部分组线性无关．
2
（单项选择）在下面的支付矩阵（表10-1）中，第一个数表示[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的支付水平，第二个数表示[tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]的支付水平，[tex=2.786x1.214]Alvty5eRAguf/BAip0SU2g==[/tex]是正的常数。如果[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]选择“下”而[tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]选择“右”，那么：[img=966x165]17b260e76ac06cc.png[/img]（1）[tex=2.286x1.071]/1mY4Ps0DpW+Qvb9So7bLA==[/tex]且[tex=2.357x1.071]wQTM8GfGiwurNccE25vM/Q==[/tex]（2）[tex=5.357x1.286]y8oGMJdKRfg12wyMvAGf9A==[/tex]（3）[tex=5.429x1.286]khChX+rq3BeylPDdjzuSnQ==[/tex]（4）[tex=5.357x1.286]v0fKmKzcydIRmeI+Yl/U9A==[/tex]（5）[tex=5.571x1.286]+Y6nDGhleH41PXLDAZbrOA==[/tex]
3
说明下列说法是否正确:[br][/br]在要素[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]和[tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]的当前使用水平上，[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的边际产量是3，[tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]的边际产量是2，每单位要素[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的价格是5，[tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]的价格是4，由于[tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]是比较便宜的要素，厂商如减少[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的使用量而增加[tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]的使用量，社会会以更低的成本生产出同样多产量。
4
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级实对称矩阵，证明：[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex] 正定的充分必要条件是[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的特征多项式的根全大于零．