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设 [tex=3.714x1.214]pGlN5+7gqlWy2Pd7OhVjQXuRSVHOwkcIufsoFLsrKLFzebNfWBFD+rNXZ5tXXCVP[/tex]. 试证 [tex=0.786x1.0]O9mrDPbtU/j67mXVbT2dlg==[/tex] 相似于下面形状的准对角矩阵 [tex=8.714x1.357]NovbxKl63Ey/milqTcbe/2RjOA2RcVp2YFiB6/lCv+7ULiVaBsTZRZyQ1KyfFHpM7uV08lYmTY8BmXb4aNZWJoGgoakmVDe9qF4Dd8i8L3alCQG7RxDyOjSVg8sGqmCb[/tex],其中 [tex=1.0x1.214]mh+W6LH7a6asJvd1C/N9/w==[/tex] 或为 Jordan 块 [tex=3.5x1.357]wfiHn6lCgzFOTBRaG9+epqQh8wR82oWEuofOi+VhX8rMhWrbyuL7HcHiXw5UPmOW[/tex],或为[tex=18.357x5.214]jcCMHflCR8OS9TosV6N5vKOe+sy4vJwVKwFcFxRm67sE2qeZF2F6UKSIsHTqzSfu1ybs2GEJ6kS9EIapSFFlBigBJpWgBctx9UfcSfW3IXritbuyUqvQgUr6DwB0NQfrdb581+DOz7rpKYILVD42mmSCfewa47BbOUhUcHfx/TdAwLMifaHVHzsk8qtb1+O2bfmmqO50pcAy7MOykURm9sDfLbtNzexAKUpHXwfUmg2wOB546RtkrT97BQ+80E00o/99ma1Uyu3kz58mwteyt52MV9R1LrgohAk9Kp6mgGFBmQ8XDnDTNMDWIfevModJ[/tex],且 [tex=3.714x1.5]gpsuj1OU8BAu51tHqnbOP6oUAJsYizftaxtJShp/iUY=[/tex] 而且除 [tex=5.786x1.214]IVRfzIokhPcdE8xqf2uDjhpb55Ou8HH5rLnMI2Ic8/VYQH57yueOvkYB8YAFOjM1[/tex] 的排列次序外,上述形状的矩阵是唯一的(称为 [tex=0.786x1.0]O9mrDPbtU/j67mXVbT2dlg==[/tex] 的标准形).

设 [tex=3.714x1.214]pGlN5+7gqlWy2Pd7OhVjQXuRSVHOwkcIufsoFLsrKLFzebNfWBFD+rNXZ5tXXCVP[/tex]. 试证 [tex=0.786x1.0]O9mrDPbtU/j67mXVbT2dlg==[/tex] 相似于下面形状的准对角矩阵 [tex=8.714x1.357]NovbxKl63Ey/milqTcbe/2RjOA2RcVp2YFiB6/lCv+7ULiVaBsTZRZyQ1KyfFHpM7uV08lYmTY8BmXb4aNZWJoGgoakmVDe9qF4Dd8i8L3alCQG7RxDyOjSVg8sGqmCb[/tex],其中 [tex=1.0x1.214]mh+W6LH7a6asJvd1C/N9/w==[/tex] 或为 Jordan 块 [tex=3.5x1.357]wfiHn6lCgzFOTBRaG9+epqQh8wR82oWEuofOi+VhX8rMhWrbyuL7HcHiXw5UPmOW[/tex],或为[tex=18.357x5.214]jcCMHflCR8OS9TosV6N5vKOe+sy4vJwVKwFcFxRm67sE2qeZF2F6UKSIsHTqzSfu1ybs2GEJ6kS9EIapSFFlBigBJpWgBctx9UfcSfW3IXritbuyUqvQgUr6DwB0NQfrdb581+DOz7rpKYILVD42mmSCfewa47BbOUhUcHfx/TdAwLMifaHVHzsk8qtb1+O2bfmmqO50pcAy7MOykURm9sDfLbtNzexAKUpHXwfUmg2wOB546RtkrT97BQ+80E00o/99ma1Uyu3kz58mwteyt52MV9R1LrgohAk9Kp6mgGFBmQ8XDnDTNMDWIfevModJ[/tex],且 [tex=3.714x1.5]gpsuj1OU8BAu51tHqnbOP6oUAJsYizftaxtJShp/iUY=[/tex] 而且除 [tex=5.786x1.214]IVRfzIokhPcdE8xqf2uDjhpb55Ou8HH5rLnMI2Ic8/VYQH57yueOvkYB8YAFOjM1[/tex] 的排列次序外,上述形状的矩阵是唯一的(称为 [tex=0.786x1.0]O9mrDPbtU/j67mXVbT2dlg==[/tex] 的标准形).

发布时间:2025-08-22 19:42:46
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答案:证  因为 [tex=3.714x1.214]pGlN5+7gqlWy2Pd7OhVjQXuRSVHOwkcIufsoFLsrKLFzebNfWBFD+rNXZ5tXXCVP[/tex],故 [tex=0.786x1.0]O9mrDPbtU/j67mXVbT2dlg==[/tex] 的初等因子为如下:[tex=19.214x1.571]QHbKoSD/Tc0N8916667lcfBpI+v3vWSBSoaT4BIVd9m0AGVanKgqviRq+ctwhM969JcSdWX0gNkaHq4wU8udxyHqisOHgm/xKA3fzWT6ydeFN3jC+J54E9hyjX1nrnqH7B8ZJ01P569/pe0p3LKkphnTieuwq76DLsA1z/IwDE1Bbv3kqtlOyEbm88gAdgaCBLI2f4hcZHQ5vQJr/x9LWmF7KJ7i9KBcd/vXMBiGvXo=[/tex].而 [tex=3.5x1.357]wfiHn6lCgzFOTBRaG9+epqQh8wR82oWEuofOi+VhX8rMhWrbyuL7HcHiXw5UPmOW[/tex] 的初等因子为 [tex=3.643x1.357]e0Zl7OChjKdCCKMNNl410xg+gfQXxQHdQmD+vW9dcpKF/36ldAA8YgB5yqk54UfU[/tex];矩阵[tex=10.5x5.214]jcCMHflCR8OS9TosV6N5vKOe+sy4vJwVKwFcFxRm67sE2qeZF2F6UKSIsHTqzSfu1ybs2GEJ6kS9EIapSFFlBigBJpWgBctx9UfcSfW3IXritbuyUqvQgUr6DwB0NQfrdb581+DOz7rpKYILVD42mmSCfewa47BbOUhUcHfx/TdAwLMifaHVHzsk8qtb1+O2TVSdnasHiMSF+gDcoOvb/Q==[/tex],其中有 [tex=1.143x1.0]2V+AKfOVjCGDMcEl6c1k0w==[/tex] 个 [tex=7.5x2.786]D67Hk5bs7VCj6FbF5yUZKAC7Jkda40wM/YmVbM0ZvwCzrvkbXVwSeh/BWeSMKFftzKsmsJdwS/ak84vM9xCm/cW4xZpfTL0R3QAcxDVmbP0=[/tex]的初等因子为 [tex=6.786x1.571]pJzeTo8YJJUpX24bqHJWTWmgoWfHsPKYyQXtZZStkox+AQH5OBqH0uINYvmD6tizOIrsx4nN4uk8HVcylb/a9Q==[/tex] 因而结论成立.
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