设 [tex=0.929x1.0]FV0k2T/xaj6dPCbFnkB3/g==[/tex]和[tex=0.929x1.0]ep004cu6Ev4qhlMpamsNGg==[/tex]是两个同阶矩阵,证明以下命题设[tex=0.929x1.0]FV0k2T/xaj6dPCbFnkB3/g==[/tex]和[tex=0.929x1.0]ep004cu6Ev4qhlMpamsNGg==[/tex]是两个反对称矩阵,则[tex=1.786x1.0]kxtqona9AyNIu9S8LHQ7Cg==[/tex]为反对称矩阵当且仅当 [tex=5.357x1.143]R9q/1zIcSN2eT74WQYLXSyFIAHbmLC//siVzq2KtWJdfsczy+dr6NntkYOOK2gjp[/tex]设[tex=0.929x1.0]zkuxy59wnc0FrSuUc1OFF6pw7am5S+IP5AAfiovVsGI=[/tex]和[tex=0.929x1.0]GTnOCR9hNPsOuxGSyBGTAE4D+bwdNZdKWKqAkIkho7A=[/tex]是两个反对称矩阵,则[tex=1.571x1.0]mCjAngcIqtveplNftuY0BQ==[/tex]为反对称矩阵当且仅当[tex=5.357x1.143]R9q/1zIcSN2eT74WQYLXSyFIAHbmLC//siVzq2KtWJfBR5FBks8WxafH8Ysq8zxu[/tex][br][/br]

2022-05-31

设 [tex=0.929x1.0]FV0k2T/xaj6dPCbFnkB3/g==[/tex]和[tex=0.929x1.0]ep004cu6Ev4qhlMpamsNGg==[/tex]是两个同阶矩阵,证明以下命题设[tex=0.929x1.0]FV0k2T/xaj6dPCbFnkB3/g==[/tex]和[tex=0.929x1.0]ep004cu6Ev4qhlMpamsNGg==[/tex]是两个反对称矩阵,则[tex=1.786x1.0]kxtqona9AyNIu9S8LHQ7Cg==[/tex]为反对称矩阵当且仅当 [tex=5.357x1.143]R9q/1zIcSN2eT74WQYLXSyFIAHbmLC//siVzq2KtWJdfsczy+dr6NntkYOOK2gjp[/tex]设[tex=0.929x1.0]zkuxy59wnc0FrSuUc1OFF6pw7am5S+IP5AAfiovVsGI=[/tex]和[tex=0.929x1.0]GTnOCR9hNPsOuxGSyBGTAE4D+bwdNZdKWKqAkIkho7A=[/tex]是两个反对称矩阵,则[tex=1.571x1.0]mCjAngcIqtveplNftuY0BQ==[/tex]为反对称矩阵当且仅当[tex=5.357x1.143]R9q/1zIcSN2eT74WQYLXSyFIAHbmLC//siVzq2KtWJfBR5FBks8WxafH8Ysq8zxu[/tex][br][/br]

答案：

证明   设[tex=8.357x1.429]c5Cf4pRARaBipYntugL/3hfqyKzP+4z6x7UtzURsK++YRMMrziRwxC0jmgUUccdECzmteE/fDvgz60vhKc+GLbBOzKe2g0m99fhYlOjvszu6tDa591QYwajRnAVWezrV[/tex]则 [tex=14.714x1.5]Pl82IheWxVFVdJ2+Fd7sovKvigXNNumVpspBZyZ/nw01/ZS2oxOafpE6m8YSWMvSJrC2xB5qGx+h+ghtxVhDhfS5UfOsrR4trcFPbSDzeC8QhjXKSmzj3WH8tTFw/mBsijUaU5TJyiqwxsNb3yUFHnBZITa04Yh6XeFssDkBPuifD8xV62arh6KYRmN7hezwtlmA3WVV31ObLNMEnjEOtw==[/tex]. 所以，[br][/br][tex=1.786x1.0]96v/QovsZC+SUffULwqQnvkECUDsFAQTkNLNhDaRrpw=[/tex]是对称矩阵[tex=12.714x1.5]bMRrINhuwlMbjrHDeWypotqyWOgtqPxdg0MM6jJUselzYnQw0gOLRe9DAsahoPXfam3btW10R6/IcK3Y4sksxG0vd5QmF0IBjJQ5gL4rtsmcVxaOeLyN9O3RJCnui23aIar2BBYIefzMLr/f9qCkCVj3dlB99bVPrJaoVgxyO0Y=[/tex];[tex=1.786x1.0]96v/QovsZC+SUffULwqQnvkECUDsFAQTkNLNhDaRrpw=[/tex]是反对称矩阵[tex=14.143x1.5]bMRrINhuwlMbjrHDeWypotqyWOgtqPxdg0MM6jJUsenrIkIuzQH0dC69RhPoGsgoTG0Isaro9djgiWyIedCWRBnx14MgV+lwDvjlP+yh7vsSvpKSpDTWyEqurmsfrjs2Dsltb4jDtYWGdcyYUXqkbBZsxWKP67n7b45alL0fFWQ=[/tex]

举一反三

内容

0
设3阶矩阵[tex=0.929x1.0]FV0k2T/xaj6dPCbFnkB3/g==[/tex]与矩阵[tex=8.286x3.643]MVsnGUDjteIUZsyQP8wk5vVdn9LVel+idc+e1Za3dgyv9nteLYWCA59rqAXQh7A5N1Tx1TEMHtNa3k21UEOh4ZbsqFh2sA1PN9AvHMj9QjYyvsedS6KXBVVUc1yGZpg+[/tex]相似，试求矩阵[tex=0.929x1.0]FV0k2T/xaj6dPCbFnkB3/g==[/tex]的特征值．
1
令[tex=0.929x1.0]FV0k2T/xaj6dPCbFnkB3/g==[/tex]和[tex=0.929x1.0]ep004cu6Ev4qhlMpamsNGg==[/tex]为两个[tex=2.429x1.071]kaIcCzgC6SpeVVzRje1dYA==[/tex]矩阵。证明：[tex=8.214x1.5]95l45PjlG6z6EXOMXiCKI0S6lKuUdYJpOgS3I6rNDUiY9B03ZOC2T/ProZwaNfOpPwM6VzjHwR4pICa3sb2gz3yEhlkJp01Yq1Il+Jv9rXcUiR+smmYbCU7k+kOwhM+TI4wXRX/cYFIZKnBr/6cddA==[/tex]
2
设 [tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex] 是 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 级实对称矩阵. 证明 : 存在实对称矩阵 [tex=0.929x1.0]ep004cu6Ev4qhlMpamsNGg==[/tex]使得 [tex=2.929x1.214]qLfCK1ZvSHsu4VEM0GGu98UJear4tHjmNm3vBZGGTAheeWeDVf2rrdw/E7PJySLb[/tex]的充分必要条件是, [tex=0.929x1.0]FV0k2T/xaj6dPCbFnkB3/g==[/tex] 为半正定矩阵.
3
设[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]阶矩阵[tex=0.929x1.0]FV0k2T/xaj6dPCbFnkB3/g==[/tex]及[tex=0.5x0.786]ICKY+F5VdoSQrRn/wUUOyw==[/tex]阶矩阵[tex=0.929x1.0]ep004cu6Ev4qhlMpamsNGg==[/tex]都可逆，试求：(1)[tex=6.143x2.929]075gCzZzsMRb6HYXYk9X96c14hJEyYCrY/NHnamm9AzjsMfiXcz2/LWAFEj7i0qmE4UKQFEEygPspsOQb9gk05QzMYAXAEObZR9eDWlY+jnF+27pEiiSSp5A7r5Xcof9GwAsfCpQmnd909aMAabKxLNjgRTKkjY5XfD8BuJlOjM=[/tex]；(2)[tex=6.143x2.929]075gCzZzsMRb6HYXYk9X96c14hJEyYCrY/NHnamm9Axv3wj6cDXa/m0w59qdSbnGORVnLZw0ipt9bsnZC/QIfTqebN5qQ5h5IoHPHJmWEsDrKwrM6Rsd08W6HukTXbTVncEmYPN+kyS2CJL6gsQ186AcylhnhB2NkiY5RvVCjIo=[/tex]．
4
令[tex=0.929x1.0]FV0k2T/xaj6dPCbFnkB3/g==[/tex]和[tex=0.929x1.0]ep004cu6Ev4qhlMpamsNGg==[/tex]为两个[tex=2.429x1.071]kaIcCzgC6SpeVVzRje1dYA==[/tex]矩阵。证明：[tex=6.714x1.5]iZkg9HotdykN5NmHnJ/droKGin0pemAbZgZ3JDVuXyV3ElyEJ9YFaHo7qIXMz/YmgnUO1SD1CXzJYG3SsoAlJdyVw2R1tvjF7xGI2/T1z3nFVfAa0QPEO/d45aHGfcTtrAPd8dB1/HKhlQFlYPhLVA==[/tex]