证明 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶实正规方阵是两个实对称方阵之积.

2022-05-27

证明 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶实正规方阵是两个实对称方阵之积.

答案：

证 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶实正规方阵 [tex=0.786x1.0]PutU1cWdyHyySBp7YfCWhQ==[/tex] 可视为 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 维 Euclid 空间 [tex=1.357x1.071]76Y/4zW6rg9n4Kj9gi+C3PvBvnXzVcL3VBl1fvQdg7c=[/tex] 的正规变换, 于是有正交矩阵 [tex=0.786x1.286]lxpRytJLiBEazKq2/TqV7Q==[/tex],  使得 [tex=7.143x1.286]goxnaFMneWzuUgBjDzFfdM8L/H03jSMnoyETsvTMqroHE9UP0CHvuZ/Mm0g+FhNA[/tex] 为标准形: [tex=17.5x1.429]B/vLqgcX8TZCQh98PwxZD25xGKf3xvb7C1vmkCvn29VlpIrWtnNF0hL18NUsQ5qkTKKhbtEEnZX85+DaMmtyqbJ1LYdJbwhWwPV7wNVWxNOFCxFlhtPbpkL4+H+2e351Gszm8JFty/hQgskj2hJ7uO8P2BMiZGc8MQKNTNV9H01ED6/DyMJZHwHN79A2a7Py[/tex],  其中[p=align:center][tex=19.857x2.786]JH7XeIrTk7dyu/mJNFxicYqY/2m73lBWNuIT6Bn0uPnOztTcQEAlWn6GeynVjGP9Mo19WEUeZSCv+BjCrb9ZFWcA/wgVqleo6xOMUXIaEDO+rmSBWIgNFKmj5x485D92Eb3KYlE4DQXnS31bLM9s/uY0NPdcD0ftE/rRJjgqRSw0E9nWcnD0vez6E9mXiQ8zccy2X6q413yfnui1yrieX1raCvZu0TqvNfygyJEi7Mo=[/tex].[p=align:center]注意[p=align:center][tex=6.357x2.786]lBzc0rtB7NSrO7NgxXLJ0oZoV8bz122ILs/Z2zx1blqx9vyAtDOHQoJc//G6+GZAnGwBuvzKnov3ERRe9j2+ku1AhO8uIeotogp0Jyj5+gU=[/tex],  [p=align:center][tex=24.0x2.786]jcCMHflCR8OS9TosV6N5vJPKTza1UrQuRfSxrADuqhfhR1CllkvcLB57Xm1Xsz6ecNJptwXPdkG5Ocs50xLVbLBhwNO15yWSW2yM7OYduRRTudCkchI9pYh1sibXfoD8GsXbobpvmssCdQD2VCIuQ1PDde+U9OeWinfE/GIYG8xwAMgxv+pNhxVE9nFqsef+3zjv5kXwc9MUSImOwYCfd/+c7dySbyipNWcKNaEh6/kHY1LB08e+rz+3jDD2+UIonlG05dBmQG1hRODUy7zHjiWBUL5vg5Go4vrtfj5L4JRTLMtKfKmGbzgAnO9PzHQtaOMAjhDVS1fqvEGsPvVBDwSkkJW/s7/ftFT3pAvaaIkZ/YeZv/vKnu57VrMAIAQDBqTO/V8Zx/M9pyXDwSmzbg==[/tex]都是对称矩阵. 令  [tex=11.429x2.857]6JY7nlSVf4ozpOCZca6YmqZ58zisEGKAQhiev5CAS7OlDJkowHSHHH9ZI1X2x6wyQQWPnxX5GsC+DeaQiYZM0vyzlND0w9AZAD1iyEPgei4P4+etfHPB37hJXD+g1yY9[/tex], 于是 [tex=4.357x1.286]KAR4Auq0zLlsVpzUg8BG+bXBPPLjbhupVAfY7VCvCEYdER21uQMqNfEqKT3M56Uz[/tex] 是对称的, 进而 [tex=8.143x1.429]bYVnhD4dBptltH0mjvbHP6KrllQz/qyq0SRvZ/wC3Z8FXmV5lxRgVtMIAv8sks9akKFMaCTZn1xGu6RmPpMVU/E5M8G3JaYrK83c++3ew34=[/tex] 都是对称的. 再由 [tex=7.714x1.429]B/vLqgcX8TZCQh98PwxZD9zyy5BaI6Tf8IRCPAH4wyeL06JgghTaJ/eLNUX7mnPzc5OLBtqHzKX6WJcNDDx/Wg==[/tex],  于是 [tex=11.286x1.429]wM/lU1m6vXzFRyWGNdIQ24Xp4aCuSwW5AhY0nQqRHYXWQedoIc3DybvxAAc5Pkv5b0S/cMmqs8nZlhe5Ev5/HRd09Ipwx46XFunkK9aXa7bFOt1QMGV98jTjw57SHPXr7OXujA8Ko7guC4ybKXWzKA==[/tex] 是两个对称矩阵之积.

举一反三

内容

0
[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶实对称矩阵 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 的特征值均为正数的充要条件是 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 为正定方阵.
1
若[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶方阵满足[tex=3.0x1.429]T0Dzim7yNRK+xgSgJZLNCA==[/tex] 证明:[tex=8.286x1.357]B6dUbBy5O5FR8WSysk+6jg==[/tex]
2
设[tex=0.643x1.0]hK6dRoCn+OGpoJ7dSqNW4g==[/tex]是实数，[tex=0.714x1.0]J/aA9EEo0KmJFnWWfX7LmQ==[/tex]是[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶实方阵，且[tex=4.071x1.214]rgVkZ9NqZTWmPUaTfd5Vnw==[/tex]满足[tex=5.929x1.214]C7uLfPpkC5aMiIavayLq/JCMUBp5cjQMhu0f/HwbwAo=[/tex]，其中[tex=2.5x1.357]KGZ/p+EUV3T2O1QX4dN24A==[/tex]，证明[tex=3.571x1.357]xaEvko2GK0gSU2NNJ3ZV7w==[/tex]且若[tex=4.286x1.0]uqh+oOvD2P9iqZ7dD7XO1OkkgjGw/1cC8lDdj9Z9gaI=[/tex]则[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]是偶数且存在[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶实正交方阵[tex=0.786x1.0]YEkxBRWVe8SyiK/VG6WTCQ==[/tex]：[tex=30.643x4.286]MWTlCxo68JHY04c3sZlDTsjd6uNnqckz5babdKdcZJKguas7nrJHJtors1evuUamtpZ+UHQGoYcx+4ple7KNUwHKjL8Az2kNx4JfEMzb9tfdfNUxw8IM/J2zfEIKMkWCdWETzhke0ysDSnP66vL4EGI92ewR8ZSgeZfi69FZZaidFS4ypW2TLbObUhZQqH9DHIvPonLr6NopqEI92jvUZd5bV/qrC+vK0eJ8iYF5Tp1T00zhgmOH3Pf0kRmbdovWStBG60zYSrFwd+0NoVXDxg==[/tex].
3
设 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 是 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶方阵，证明: 存在 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶非零方阵 [tex=0.786x1.0]sHo1pKm+gjxjcUAJjHrarQ==[/tex] 使 [tex=2.786x1.0]I+N8DyvEXHfXhQN/cOpe/g==[/tex] 的充分必要条件是[tex=2.643x1.357]1u3XhOXVwmW3C2B6QBCBLQ==[/tex]
4
适合[tex=2.786x1.214]5YJ7IJv26przrQ/Z5urQMQ==[/tex]的方阵称为对称方阵。证明：[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶对称方阵[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]和[tex=0.786x1.0]sHo1pKm+gjxjcUAJjHrarQ==[/tex]的乘积[tex=1.571x1.0]mCjAngcIqtveplNftuY0BQ==[/tex]为对称方阵的充分必要条件是，方阵[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]和[tex=0.786x1.0]sHo1pKm+gjxjcUAJjHrarQ==[/tex]可交换。