• 2022-06-08
    判断矩阵[tex=7.643x3.643]jyVOORWehIbTNQvvtYroWlzdxT+GgGCAXLAVyqZsgVpJUuQD9vunAJguTgz65pUM6A2Ttd8uTrC4ww4v79AVwz6DiygFPO3JAj1F04/3E75E3gAfyap35Dj6OfEka8Gz[/tex]是否与对角矩阵相似;若与对角矩阵相似,求一个可逆矩阵[tex=0.786x1.286]dSWbQCTjdbLxKy7q0ps2gg==[/tex] ,使[tex=3.357x1.286]QehgMsIi+Hsdet9OihqiWQ==[/tex]为对角矩阵。
  • 解 [tex=0.786x1.286]pi/GsQ3apuRt43V3XQq/tA==[/tex]的特征多项式[tex=22.143x3.929]vAGON7+JxTsi8RkBcCBNjOdWSD1avKTKahYRgaf+lGVCp8jfZnhVO67+mdxK4qArlZbBQflvXT6wkKR6Eu6Y9a6E9bse1A55CQF3PlGHKTLT8GJNFszFEqxIqyY0tR1bt+kvRz++TOXXP9XsfIWr9xXkcCsSysy5oN93MpbsqbhrWFgEwDrnE8ojZsobIsMCPGFWYxOXbFynSGUUdyH/L9DirhKYak4eM/spxnjzXBE=[/tex],[br][/br]则[tex=0.786x1.286]pi/GsQ3apuRt43V3XQq/tA==[/tex]的特征值为[tex=3.357x1.286]JjUvEVviPupZHWIJEKQ81JSo6mehDs7/DSxPf+ijG3c=[/tex],[tex=3.571x1.286]uM9AVBm+eqrBQEgobwY5d//zcIMBf+L3ey1hqraacos=[/tex]。当[tex=3.357x1.286]JjUvEVviPupZHWIJEKQ81JSo6mehDs7/DSxPf+ijG3c=[/tex]时,解方程组[tex=6.143x1.286]gUVUR1Dp4YjRzmJpiUN8ig==[/tex] 。 由[tex=18.5x3.643]9pIEGnGn+4yjElCG+gl8JjQpUH9RDE4ayD0pNBVNpaFsIFhUrRT+CTjHxD8406I8tgvD+q7m91s5veYrB8iS3qzIwrFVzLOh3stKN8wTHWVEMQ9gztmcTiKkne7IxEI2/GEZJBXGQ2cNdlsHBn8l/BW9sAO1Cggy/4ywVaOOgcNcf/iiHYrE/1oVk7Hc0cgrVDj2jtN0zEmoolNbd8wTVThf5KnJCdyhrHeivpHBWIIEgyCTPfNVrCl/M7uCScv2uTlcjJbzqSLRBaplIWidbA==[/tex],得[tex=5.929x1.286]DgPgGXkn8Kl8PzK4SO68ug==[/tex], 所以[tex=0.786x1.286]pi/GsQ3apuRt43V3XQq/tA==[/tex]与对角矩阵相似,且 [tex=4.5x1.286]WPrBWmdvXNEuZCYOwyxKvpL+hdaTCbMNrbx72wQJ4Kc=[/tex]。 令[tex=7.214x2.786]rwMhqGKFQ+j3l2qMx/grPt9qN706kR0SOzgund7Ytv2NQMR9B2zApiRIZcHdR6Ae4+nGp0u+VFrZuoFWVoiFRzPXnYY1O7Z5MGwq5XEFuL5bVXs4/O7/njD6VLrmbJBobdXuD66TF4FwlPUq5NdLIw==[/tex],[tex=3.0x2.786]jyVOORWehIbTNQvvtYroWmC1E5bhuHRN3BGpXiLYR7NTaWIr3SZgFGGmHyJHkIdc6ZJJNvbMQEn/RyeEi+Mq8A==[/tex], 得属于特征值[tex=3.357x1.286]JjUvEVviPupZHWIJEKQ81JSo6mehDs7/DSxPf+ijG3c=[/tex]的线性无关的特征向量为[tex=7.786x1.286]tMXJSwqqism+HNu6MyX1krXIbUPBkUzhjuo7kXI0+8I=[/tex],[tex=7.0x1.286]mmxoEhHutq9KTSc4HAI5U8EcH0huWzK6bV6IUNCRHsY=[/tex]。当[tex=3.571x1.286]uM9AVBm+eqrBQEgobwY5d//zcIMBf+L3ey1hqraacos=[/tex]时,解方程组[tex=6.643x1.286]Q1E0Ektg2t/drqqzYiMFRv4YRTmpnRmRJm+PKcR4gi8=[/tex]。 由[tex=19.786x3.643]Ab3WB6TS4a0JVVv4S4DPZG/v+vuL2VAcRBmqMv2BMAgHUD6BSxVx7CcGPRbbSTZr9mro2s+oh5Xqsn1yc9vhZtY62XK9D2H9sTSuRAdpFniUUPZNWyJuYWvYst66SYQCKSfpP4E4HBS/I+zH9CUs+WjenhJquMnCKZIRNSer6AyyXSpbfaoXcSy/UsuSi5ePnTMs5CABC+ghguEsV3GpbD4dPuFJ1pGoTz3qV20Wa3zDS7aP70I0L/EI9Evi8fM3HHg312i5a4lU3Jpf5B8W6Q==[/tex],得[tex=5.214x2.786]M/Yeox5bOq02SPK7XRukb2XouFUv5tiTdIpz1KI9BLS1ND/xSe+3Sx+GYrpm4xuy0WFbAvgQdHhjv5dzwbklTG8qQcNFNKR4kSbn3TTp7Sc=[/tex]。 令[tex=2.786x1.286]V2s+IyDToPuLJq3zGNXxzQ==[/tex], 得属于特征值[tex=3.5x1.286]YMmdZEPmcSQzuhFq9M+TKo/Yh5MGB3dkNyDLP5TQ4uM=[/tex] 的线性无关的特征向量为[tex=7.786x1.286]XAMS2LFHnC/9x0IbRf79SiMqKqu//oMmBsoqDSHLcro=[/tex]。令[tex=14.286x3.643]Zh6eOjHYxWysFAM95QqAPx+QxZMl3+gE7b+jmOFbzU33dkJj6xqknorPvL9AVvkuAuBtv3zv3pSS8y5XeZfnl1vWUwjNf+8/FeBD+XOvKSTyUM3AqhNz8xFFkEr9A6tgwXW3Mr7MYuBcjxfkatUtc4KfvLSpmJSMZh3UcOs4BXQ=[/tex], 则 [tex=12.429x3.643]rhGl7XA/qs6+9yphb1bfqoAurp+0lIZrtCoitmbPWN/gspW3u6CK2KWTotfc3snkSCMK46UfOlm1Vf+WVwgAsjSeq3tzMMiGKNMvNjv5Iv2111liOZsGTZwyRFkDY3fvfD9qi2mfc3suaHhaG2GCFw==[/tex]。

    举一反三

    内容

    • 0

      下列矩阵是否与对角矩阵相似?若相似于对角矩阵, 求 [tex=0.714x1.286]BMKsEVFNvpiLV0UsqDFXCw==[/tex] 使 [tex=3.214x1.286]1sFdXzzqHCc4qLXdM8ki4Toicy90dsQBVNzF4LK8JX8=[/tex] 为对角矩阵.[p=align:center][tex=4.5x2.786]jcCMHflCR8OS9TosV6N5vEh2mkwOQPgJaMjhXUUvt/u/aDMg7Joull2yzyzjRLUkka8hhHVAcdjyppjElztjEA==[/tex]

    • 1

      下列矩阵是否与对角矩阵相似?若相似于对角矩阵, 求 [tex=0.714x1.286]BMKsEVFNvpiLV0UsqDFXCw==[/tex] 使 [tex=3.214x1.286]1sFdXzzqHCc4qLXdM8ki4Toicy90dsQBVNzF4LK8JX8=[/tex] 为对角矩阵.[p=align:center][tex=6.929x3.643]jcCMHflCR8OS9TosV6N5vAsRiX0HRDF6deVm1P3ZNbcryL/4z1YRStI9T1RLH8oBSlyEGqOTzkzDCbMq0bIIXVYueNkOeT26F02GTHQKQUW3/9zfT6cXZLWR2NMKEfA3[/tex]

    • 2

      下列矩阵是否与对角矩阵相似?若相似于对角矩阵, 求 [tex=0.714x1.286]BMKsEVFNvpiLV0UsqDFXCw==[/tex] 使 [tex=3.214x1.286]1sFdXzzqHCc4qLXdM8ki4Toicy90dsQBVNzF4LK8JX8=[/tex] 为对角矩阵.[p=align:center][tex=6.143x3.5]jcCMHflCR8OS9TosV6N5vBd033QJVVKN8Pu/Wce/RY+4F9/ZEYCNLpGGCW95mbfQVIkZ3V/nk34FVk/TFVprQr5n0GJXqwW3N/OKYy3Z7PY=[/tex]

    • 3

      对下列实对称矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex] 求正交矩阵[tex=0.786x1.286]dSWbQCTjdbLxKy7q0ps2gg==[/tex], 使得[tex=3.357x1.286]TvRsUv5t/3lJOPEdcGcONfuVM4C/bLaP8ZJtWRinRXA=[/tex]为对角矩阵:[tex=6.214x2.786]sSXBpxJWudVpH1R35o4LnE0bLZnSPEoDNmtl5XLvZQ81q6AbPwVhJ0ckZM/g2nUxqJrc7JTIzM2sUXDRpC7mKQ==[/tex]

    • 4

      如果矩阵[tex=9.357x3.643]3BT1BgBZQ5uJXxD5dg+w25oxXRH9+KEjMiNSxk6AZG+PsFZwXRxPIBN6s44j902W5vNNmOjVBXquMCKEgf/BNJ5SSXfj6kONTH2cuTJkBPIBOYobCdzUTg1N8KvXoIB3[/tex]与对角矩阵相似,则写出相似对角矩阵[tex=0.929x1.0]tfHTq5YB6eq76zMfF9z56Q==[/tex] 及 [tex=0.786x1.0]6J6pLBwELDvuZYB9vl6pdg==[/tex].解: