证明：如果 [tex=0.929x1.0]JkZEjSnuwtkZlFnZMXvQ5Q==[/tex]为正定矩阵，则[tex=0.929x1.0]JkZEjSnuwtkZlFnZMXvQ5Q==[/tex] 的伴随矩阵也是正定矩阵。

2022-06-16

证明：如果 [tex=0.929x1.0]JkZEjSnuwtkZlFnZMXvQ5Q==[/tex]为正定矩阵，则[tex=0.929x1.0]JkZEjSnuwtkZlFnZMXvQ5Q==[/tex] 的伴随矩阵也是正定矩阵。

答案：

[b]证法一[/b]   先证[tex=0.929x1.0]zkuxy59wnc0FrSuUc1OFF6pw7am5S+IP5AAfiovVsGI=[/tex]的伴随矩阵 [tex=1.286x1.071]c5Cf4pRARaBipYntugL/3vKeBzcFZmpil4mkUJnj1jI=[/tex] 仍是对称矩阵。因为 [tex=0.929x1.0]zkuxy59wnc0FrSuUc1OFF6pw7am5S+IP5AAfiovVsGI=[/tex] 是正定矩阵，则 [tex=6.786x1.5]wTJjhoO/FrQzcnhAzbBA+urLz/9zk4kkFJcmJh04n4enZ7MyoIB3JwiU89Q52mmc[/tex]，所以 [tex=0.929x1.0]zkuxy59wnc0FrSuUc1OFF6pw7am5S+IP5AAfiovVsGI=[/tex] 可逆.。于是 [tex=1.286x1.071]c5Cf4pRARaBipYntugL/3vKeBzcFZmpil4mkUJnj1jI=[/tex]也可逆。由[tex=0.929x1.0]zkuxy59wnc0FrSuUc1OFF6pw7am5S+IP5AAfiovVsGI=[/tex][tex=1.286x1.071]c5Cf4pRARaBipYntugL/3vKeBzcFZmpil4mkUJnj1jI=[/tex][tex=3.071x1.357]iQr2U3D/6iNedKuJfDhRi0CeMUs+eCl9THvaYiXcj7wgS9p/FJACr5BiBS5H1yb4[/tex]，可得[tex=1.286x1.071]c5Cf4pRARaBipYntugL/3vKeBzcFZmpil4mkUJnj1jI=[/tex][tex=4.071x1.5]iQr2U3D/6iNedKuJfDhRi4yHrnwWSKRQbD52PW/P055W8wINMdiRf5vZMsGVHVdi[/tex]，所以[tex=26.214x1.786]v6h0emw+MX5fBgkKQJsSsI8skzOJQShvO+Y0b+BDva+HNvue2R9ctoYSQZHK1iaAo4bPN8Qku3fsJGko2SePEyJmVkpW3eMsH7dENTmusnGQmtS1fw+m6gRBtLWrLVP8e5TpC8WBmZ6EDyWaqdbYZ+Bs9KIZbUQuP72HJyVpQLOq52XNXGkaFG9t1VmCqdo2soe7d9sqYHGKwSjweJUjx/HJAV96Ay1SBdlu5MYaM47nflqd0eIt7KVqCj4HRgBHEGbLOMI2y04Edwd0cU+/RJ1Vjy/KYjuN/sr+z23S9Ww9aunBpJwmmJwnM1paQPhMyN3WGxRGinaCp/oAqc1DBfJUFQhkQ0nzokbhIjZRW+AWr2F/c1U29S+yP1eLAtNlk57d7YXdYEZugA/2BSd2jw==[/tex]即 [tex=1.286x1.071]c5Cf4pRARaBipYntugL/3vKeBzcFZmpil4mkUJnj1jI=[/tex]仍为对称矩阵。又正定矩阵 [tex=0.929x1.0]zkuxy59wnc0FrSuUc1OFF6pw7am5S+IP5AAfiovVsGI=[/tex] 的所有特征值 [tex=9.071x1.357]LHK2eexfYo7SNefzTIGvr/aF4oi0qzdXqFp12Lor5C1CcOSugsLxpR9I4VH8zBZO[/tex]，故矩阵[tex=1.286x1.071]c5Cf4pRARaBipYntugL/3vKeBzcFZmpil4mkUJnj1jI=[/tex]的特征值[tex=10.929x2.5]EVuOuMg02rjgsgt21rWl8zP/1aIdSn3VGZtGMmZ7lv1WzdqV+DEfZM34eR2TJG0eu2S2Y0Lrb4e4g3KAc8ppRw==[/tex] ，可见 [tex=1.286x1.071]c5Cf4pRARaBipYntugL/3vKeBzcFZmpil4mkUJnj1jI=[/tex]仍为正定矩阵。[b]证法二   [/b][tex=1.286x1.071]c5Cf4pRARaBipYntugL/3vKeBzcFZmpil4mkUJnj1jI=[/tex]为对称矩阵 ( 证明同上)， 又 [tex=0.929x1.0]zkuxy59wnc0FrSuUc1OFF6pw7am5S+IP5AAfiovVsGI=[/tex]是正定矩阵， [tex=1.857x1.214]8Z99eWZ0koE5mYUHRlJvaA==[/tex]仍是正定矩阵，而[tex=5.857x2.643]c5Cf4pRARaBipYntugL/3qkGs7cYPMyMpantWL86z78UNSDHN8RDrmoX8odBOGItTEvApOsBWn03nsz1m3LJyYmaRKLfr5WD9mMlzZSrG4o=[/tex]，所以对任意的[tex=10.571x1.571]r23SDxXRKB//AckLIeV3GJwoaWOUT/4pniJ3SOhXWGSqJM86/savUovhGHIfUr7X0Gabgd2DRq2UEkM1EYARxtKqbSy5+DAgFa/y3eTsJDho/Zsf0omaTJ+6ZqUroSvS[/tex]，[tex=17.857x2.643]NntPXTpKm7a85+byh4gr6db+oPYwpuQE7lYkLo6HW8rQY4w1x7EJCIAFJ3PEnELPmsJ5NA/5nLN+rv+ViT6ko2ka5n6dQoH+HPXG15CUadj+mTHd7ZFPGc6sngVtg7+LpMZ9OmQtOGaXvsfQRZ2Lwd3Ir9jo9cErk0bPbWWomh4tiZ+rFPel33Pl16hyWjTn/fM3m2Q2mlSw2fi3GMQy2Cnv36aDQMDNHHLu8TNmGZL3MdrJuIKA0GucOv9NdJBbVZeEutCgaaxeBR31qdIy5H1vQpKteCCnPDX2qQLYcFmcZyVrQxrqzs0ILl3baVcoNVPNSQAKfUY8Fm3h4CRj2g==[/tex]即[tex=1.286x1.071]c5Cf4pRARaBipYntugL/3vKeBzcFZmpil4mkUJnj1jI=[/tex]为正定矩阵。[b]小结[/b]    一般地，若 [tex=0.929x1.0]zkuxy59wnc0FrSuUc1OFF6pw7am5S+IP5AAfiovVsGI=[/tex]为 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶正定矩阵，则 [tex=5.357x1.429]c5Cf4pRARaBipYntugL/3pDU/VqMf5UN9fHUIJ++nZ1cW2piqhtIE2I5lGhP/t+ELq847fDzLiJNnkdPo7UYLZJJ9XUEfvK+uHPvntaqELA=[/tex] 和 [tex=2.214x1.5]c5Cf4pRARaBipYntugL/3hwurkGLImGicyLqNupY81k=[/tex] 为正整数 )仍为同阶正定矩阵。以后解题时，可直接应用这一结论。

举一反三

内容

0
证明:若 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 为正定矩阵，则其伴随矩阵 [tex=1.143x1.071]nnt6woQbTr+wrutPzAntHg==[/tex]也是正定矩阵.
1
证明：若A为正定矩阵，则其伴随矩阵[tex=1.286x1.071]317mMb/UfJBjZHDU7raSng0LJvrihprruusK4SVz/+w=[/tex]也是正定矩阵.
2
设[tex=0.929x1.0]JkZEjSnuwtkZlFnZMXvQ5Q==[/tex]为[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶矩阵，且[tex=2.929x1.214]c5Cf4pRARaBipYntugL/3g4G9yaUH0tIlHD2joA/k+TPmGHmlTvDsnXlRU6g7apq[/tex]，则未知类型：{'options': ['[tex=0.929x1.0]JkZEjSnuwtkZlFnZMXvQ5Q==[/tex]至少有一个非零特征值', '[tex=0.929x1.0]JkZEjSnuwtkZlFnZMXvQ5Q==[/tex]的特征值全为零', '[tex=0.929x1.0]JkZEjSnuwtkZlFnZMXvQ5Q==[/tex]有n个线性无关的特征向量', '\xa0[tex=2.643x1.0]meAuMcYDAVZYWlVzqdNkgw==[/tex]'], 'type': 102}
3
设A是正定矩阵，证明A的伴随矩阵[tex=1.143x1.286]5WX0zEPSvFFLZ40WpRWDWQ==[/tex]也是正定矩阵。
4
设 [tex=0.929x1.0]JkZEjSnuwtkZlFnZMXvQ5Q==[/tex]为 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]阶对称矩阵，则 [tex=0.929x1.0]JkZEjSnuwtkZlFnZMXvQ5Q==[/tex] 为正定矩阵的充分必要条件是未知类型：{'options': ['存在[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶矩阵 [tex=0.857x1.0]0VpJS0vjPV56/khQ++mGPg2qyuprt2n1PFYmiqwMaHc=[/tex]，使 [tex=3.857x1.214]+HNIZcMaSzNwCe0LO7bsUnY57Z10LY7x6ACSoFezY1s4J4xYSPvR7+1TFnI90Ki4WNAhomWfwAwU9opiyejoQQ==[/tex]。', '[tex=0.929x1.0]JkZEjSnuwtkZlFnZMXvQ5Q==[/tex]的行列式 [tex=3.286x1.357]MzmmROCjjtWxSw9nY2Sa7KeGotcjG93wo6ZDcbx/000=[/tex]。', '对任意的 [tex=18.357x1.571]r23SDxXRKB//AckLIeV3GJwoaWOUT/4pniJ3SOhXWGSqJM86/savUovhGHIfUr7X0Gabgd2DRq2UEkM1EYARxpPs7egOwHj8uUXe5dYMSsMVUk8yI2kGHWY2ECfzY5LUaC3qgDC0q9hgOw21oCH9xA==[/tex]，有 [tex=4.571x1.286]NntPXTpKm7a85+byh4gr6db+oPYwpuQE7lYkLo6HW8ozxirh7BSTrxp9tlbCcGzVKb1Q4p8RDnXCrEW7VTCd9g==[/tex]。', '存在正交矩阵 [tex=0.929x1.214]BtAEkMzOBV+w/4VwoqSJYw==[/tex]，使得[tex=12.357x1.5]HhEzjKg1oPBRXjGQMpmCr9h0wnLAKzYcyA5vzedhQtYvyaDtkTJuFn7FdnlnkhkFF9/aSHTS4Wwqk2pXnnrrBRb2giN3Ar4IxXpUhEU6VaoGcA1APsJaG2uKR99KepiyTtNzsZasgJw3HpQL/6KOcA==[/tex] ，其中 [tex=5.0x1.357]LHK2eexfYo7SNefzTIGvr1TKLh9t3C5FktiIhb4YYpg=[/tex][tex=4.0x1.357]1N05TD2TAZClHDXRyewe3Q==[/tex]。'], 'type': 102}