设矩阵[tex=0.929x1.0]i0+/ldt3YBDesHpVDnGYNA==[/tex]=[tex=6.143x3.5]jcCMHflCR8OS9TosV6N5vEN9zK3ymlxcCM1AlkizbdQZOKn9KB0F2cLpBidtmh+ms5gokIdf+ILIct3hr/F6VfyPzksGlYKDTgd7xlo4gJo=[/tex]，矩阵[tex=5.929x1.5]Zb98tHKQ55nuQGi1MR0+voJxobxRnFLlswdkmOvqNvw=[/tex]，其中[tex=0.571x1.0]rFc/sfAAuCOtzhevhoREeA==[/tex]为实数，求对角矩阵[tex=0.857x1.0]KkU2h75atFwNPbzK9MCnGw==[/tex]，使[tex=0.929x1.0]tyBXjkM4oPSZ1Sowfqs4Mw==[/tex]与[tex=0.857x1.0]KkU2h75atFwNPbzK9MCnGw==[/tex]相似。并求[tex=0.571x1.0]rFc/sfAAuCOtzhevhoREeA==[/tex]为何值时，[tex=0.929x1.0]tyBXjkM4oPSZ1Sowfqs4Mw==[/tex]为正定矩阵。

2022-06-29

设矩阵[tex=0.929x1.0]i0+/ldt3YBDesHpVDnGYNA==[/tex]=[tex=6.143x3.5]jcCMHflCR8OS9TosV6N5vEN9zK3ymlxcCM1AlkizbdQZOKn9KB0F2cLpBidtmh+ms5gokIdf+ILIct3hr/F6VfyPzksGlYKDTgd7xlo4gJo=[/tex]，矩阵[tex=5.929x1.5]Zb98tHKQ55nuQGi1MR0+voJxobxRnFLlswdkmOvqNvw=[/tex]，其中[tex=0.571x1.0]rFc/sfAAuCOtzhevhoREeA==[/tex]为实数，求对角矩阵[tex=0.857x1.0]KkU2h75atFwNPbzK9MCnGw==[/tex]，使[tex=0.929x1.0]tyBXjkM4oPSZ1Sowfqs4Mw==[/tex]与[tex=0.857x1.0]KkU2h75atFwNPbzK9MCnGw==[/tex]相似。并求[tex=0.571x1.0]rFc/sfAAuCOtzhevhoREeA==[/tex]为何值时，[tex=0.929x1.0]tyBXjkM4oPSZ1Sowfqs4Mw==[/tex]为正定矩阵。

答案：

[b]解[/b] 矩阵[tex=0.929x1.0]i0+/ldt3YBDesHpVDnGYNA==[/tex]的特征多项式 [tex=18.357x3.929]Ld41YouFXqd2n+lAjQyRIdTTvT/pyghQ23x/Y1VIONO9uIEhF3HmDUtcyWZ0oIiBDSbCUt286XqfpC4WfIhkSgI8cun9xdNOnvGLrb6uedpan2AjTPf+O5rPBBUUuOohDrh6CPPiWlgFfSCSGBs4si6y0H43VU+7flb67G28XCtq2v0gAME2OT1Z8mKCm+1KMVDnVqUAwR/4d56AePmvsANwvdi4g6TAkV1ReujfURo=[/tex]令[tex=4.929x1.357]Ld41YouFXqd2n+lAjQyRIdTTvT/pyghQ23x/Y1VIONNi0/X6TQSMYpSczc4oNr8M[/tex]，得[tex=0.929x1.0]i0+/ldt3YBDesHpVDnGYNA==[/tex]的特征值[tex=6.643x1.214]6CzHpBoEVdfGanuoycm4yO8KT7B+nJSaNaXLtwG+oMRAeILMATKRFkQta12WvmXC[/tex]。记矩阵 [tex=8.0x3.5]B8Gu3cYbZ4g8279Cf9ej26QzIGqC4IR++nhQWlTpfJyggrOt9lLZkpSVzhrrIgD1iy9Y786cA3MhX6ZEibCOooWzl4hnAZDRBXJJptpDPmd7iws3JALfayY0dipoct2S[/tex]因为[tex=0.929x1.0]i0+/ldt3YBDesHpVDnGYNA==[/tex]为实对称矩阵。故存在正交矩阵[tex=0.929x1.214]aZ8kNpbAQnnvrf8BYsEIVQ==[/tex]，使得[tex=5.143x1.429]Ox/TOLjGk70cncGfDq5Kjoymjh77uVKqhn+2hjshGTE=[/tex]。所以[tex=12.5x1.786]nNAVbUQMyUQyFJ8FvVI4xBCc4u4SFA1fNFjC+NqIGR4oEifkspzeuEd09cHZFDPXh0Vg2arkNTRxIQ434CnoxA==[/tex]。于是[tex=14.714x1.786]sb0lI+O+hg9lDaI90Oub4BWmxpvXd6uFTCLn6k47PuzAg+SujlMumOtqoX+L1nxBD09WdsTRcXRwyZpKU1rxwAoAZKALdfef1x83tgyKQpkNHZrPVyl++hr52ejU54hfq7v9ond9UDuSGnuuS0ELN8i5U1+jpct0YnRHeUmzCq6ax7in2nklCAexgvOOaphL2HDde2j52TA7JzCrqbEWSA==[/tex]   =[tex=14.071x1.571]pQbquBu1TRiZU4DFRuBX7OAnbE9+M/ke1rV4XDlWT+RQSurmPiPgSJhooxpKY86y3uqdOMu8oM15id7auU9oV0CaakeCn80Uu0tPGWupwizAk0OlUVeAw/9p5TTmya/1AKcDuSJ3/Umn3thjG92DV0sDI7NZUtibM2rvVa/ET10+E3/FzzM19RpW9+8ZlOwq6UE8Hq3O1fL60TebUUxtv8zCY5Pv/BltbJYqtzlv/Y+5Ig67oZw0h5uDLTMFKBn+[/tex]   =[tex=6.5x1.5]4LTEGn5JBpJscESBQ8Fz9EseeUVCQa71iv7DndY7jkWAdTlUoKLQ6PpCPrLmlw3YjlggN6t3LWxX0D+MHcgagncGltFVtWbseBR2iB3hzP4gHBkYZbzDBrRinnssPgCg[/tex]   [tex=15.071x4.643]hupiDrrOyFSLpRkyP4SwJvhmk9li92JQb6e+0DfazVqphMC6rER0O7/7TcRguZGvG6gXCaobOuhKVQfSPTYqRq9YiaOTkLnz14nTep+cbWy2ryodEn7x4qRMGgNaj8E/9iVN/W/QFXTXbk/X8FsHmXoDjopLo2FzxkG0L6Qr3A0wPdilMuWzc8IpcmDler6v[/tex]记矩阵[tex=13.429x4.643]WNAWW+G6Rv/BWsE/cNk0R+T/EV/pnEPAeCbHn0LwyVP0maxiwICGFek00xRJV0+MjDYePlzpetlm7lRcCDPLuTZMLKiUkYx8Yx+6qlw8y9hzti++w/Uuen3pD1rWGNv6ClyDkRabNWPbF72OQhLXd+BTCREFeo3yLUnC4pkQ0B8=[/tex]则[tex=2.5x1.0]obw+x/+dSHELOSq1GMuBS5yO30Fp1gy1EesCPXTKR9Bp207aVKV8gutnDhYWR/Lg[/tex]。由上面的结果可知，当[tex=2.357x1.214]2V8NtCLstGszbDNcPZsiuQ==[/tex]且[tex=3.143x1.214]61NxqRV4052ER8BHu/1KPw==[/tex]时，[tex=0.929x1.0]GTnOCR9hNPsOuxGSyBGTAE4D+bwdNZdKWKqAkIkho7A=[/tex]的所有特征值为正数，此时，[tex=0.929x1.0]GTnOCR9hNPsOuxGSyBGTAE4D+bwdNZdKWKqAkIkho7A=[/tex]为正定矩阵。

举一反三

内容

0
设矩阵[tex=0.929x1.0]JkZEjSnuwtkZlFnZMXvQ5Q==[/tex]与[tex=0.929x1.0]tyBXjkM4oPSZ1Sowfqs4Mw==[/tex]相似。试证：[tex=4.286x1.5]c5Cf4pRARaBipYntugL/3oIqSxDEbcg1CWWvCqHxJ5WEizb5HSYlJJFpThBdYoy7[/tex]为正整数[tex=0.429x1.357]AHuNM67Mn4GAr2WKfdzp4A==[/tex]
1
计算下列矩阵的 [tex=0.571x1.0]rFc/sfAAuCOtzhevhoREeA==[/tex] 次幂, 其中 [tex=0.571x1.0]rFc/sfAAuCOtzhevhoREeA==[/tex] 为正整数:[tex=8.286x3.5]QN0fTQbn6M33pU3gx/S2ssRzVqpWUEMlPB1F8em9pxPHPIIzaitaqaXj3OkAP2YhwLgtNTq7mVpRVmzCUDjgMxeK0fRBchQXdLQiPBE6zvU4+B34aF8ZRVS24QkM3V+Y[/tex]
2
当 [tex=0.571x1.0]rFc/sfAAuCOtzhevhoREeA==[/tex] 为 ( ) 时, 矩阵 [tex=8.429x3.929]QN0fTQbn6M33pU3gx/S2srrO+uPYR1FjXG+OULT3PZ0SZZv0cSQupnZyQHcg8Bq3hHsRRoJ35sZ/ccGFpHG0e8Wn0r2X4CmeHP5XlhmRWGEElGmh4cHBsTAfC35uP2P5[/tex] 为正定阵. 未知类型：{'options': ['[tex=2.357x1.071]iILyBi8jdCgmaZqoi7cqWw==[/tex]', '[tex=2.786x1.286]Ngz0glv+VFVna1g4OsuYaQ==[/tex]', '[tex=2.357x1.071]upVaYJbqmrJHFYrKAoUxOw==[/tex]', '[tex=0.571x1.0]rFc/sfAAuCOtzhevhoREeA==[/tex] 不存在'], 'type': 102}
3
设[tex=0.643x1.286]ZsZs11iKEvfmzDIurZth8g==[/tex]阶矩阵[tex=0.929x1.0]ysPsVBYgue2sVzMz/Uq3u0A8rVtfBFebzXef4B7T0+U=[/tex]和[tex=0.929x1.0]lfAzp4E7jz98ZvbLVaeb52L/rf96Vha6tpeJ/pCuQrc=[/tex]满足[tex=5.5x1.143]ysPsVBYgue2sVzMz/Uq3uyTsebDpk7iiBhfQagFxYELnByW7YYGpbsCHsvKGvNW3ynjjf1GCTKOmGNpfywTws7aLsoJBEKNq4NdWKZItSmg=[/tex] 。（1）证明 : [tex=2.5x1.143]r5Haq7W1lVGBc4dFEM2Zk4Xcs5ubhclv3FlkYV9eqtR6YcaA5xYhbLb3ZOyZXvDU[/tex]为可逆矩阵;（2）已知矩阵[tex=8.714x3.643]k4XxnokJDFH17b6cU904x1L+ezwnamK5bBEWCJuqlAqd0xDJqZmPCLpsfN0pzpqWNK/3zauXOc34/6ExNHyRIqyz+T6tEOoZO5gdX2wOPjkuaT+XegBgVBOl93i/nYRCKhASq4FL4+S4LhdFh6VPDg==[/tex]，求矩阵[tex=0.929x1.0]ysPsVBYgue2sVzMz/Uq3u0A8rVtfBFebzXef4B7T0+U=[/tex]。
4
设矩阵[tex=8.714x3.643]5j3CprNTNJSzO282PAZoacOaMaWU2EW9sDt4r03teYs0VYQajbeeMB/nq8cJ9k8qByNQtsaAca+b+sOo4FA7K5bEm2xOv0OayWL8T7fD+hoURvEpyA/x1j1T7Be+2TXx[/tex]，已知[tex=0.929x1.0]9MCaa3NdBrky4bnBPtTtgw==[/tex]有一个特征值 2，(1) 求[tex=0.571x0.786]HXNXn3AXpwdIpZt8+6oCEw==[/tex]的值；(2) 求矩阵[tex=0.929x1.0]9MCaa3NdBrky4bnBPtTtgw==[/tex]的全部特征值和特征向量。