设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是实数域上的[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级对称矩阵，且[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的秩为[tex=3.0x1.357]jGI6hkgva7Rcyr50NnHREw==[/tex]。证明：[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的所有不等于0的[tex=0.5x0.786]Tg0I1PUwmDJ7uXa9+yiYMA==[/tex]阶主子式都同号。
证明：由于[tex=4.786x1.357]k8PvTJe4iQVkvPfhUhxDGByhTUUSYdXT25u2WyX3eKA=[/tex]，因此存在[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级可逆矩阵[tex=1.714x1.214]KU3o0cl1njWOPJN/HRu2yw==[/tex]，使[tex=8.071x2.786]GOgGvXf8fpWrP7XGIdsj82xr6/SNnYb/RmzGLoqFERyl5SHQy8OszcuP+RyB/Avmz+m/TcAlzf5mw3Gh9hjYkUNvW1jKraxW5VJIWcHRTMw=[/tex]。从而[tex=8.857x2.786]CLYI8CsKMCjgsMMrHzGKp1V076b6Ou/4X5VvxvPMh3pZ7BPg70PUjxDBEAT3JE1BdK75t4s4mt1d59LniRzCuL1ch+Vg/mlPqN0/FGO5LsM/+BJzTGmQ0BzUG8ldmJL4[/tex]。由于[tex=2.571x1.143]t2iAbYLQ2Z/Vfoa1ino1vA==[/tex]，因此[tex=14.643x2.786]KX27KyGXLj1OytPAJQvyWAjBQtgF6xzQSIzLeklE3cH++wJ5akn4eciy24FgHc+VJUUfeo8f1a5L0bKWHwiAxgzOP6p3xUAxcWYVEvEKeM7Ls+U5LxMnhl/r/vloiDLQA2GcN056Anan11kViFS+5aPpsNm0jQykPxExyn9tX+QwUiws7tnCffX5fFqKWYdjApWzjUoNo+ifR2ANHNpe2Q==[/tex]从而[tex=17.357x2.786]gA9kcq16tsV+7Wjk8w05PpXF/yrseLDPjlhfODEvZn5xTpItFuJadrAftc5CM3sFUZEQgfjTCuUIesM2p7h8OSGsFUBA+WfxTQjnHj3TKiL0S1V5odZvOwZfvOHtsu4Lzmc2kBsgdC50V/XdoisEbqUiuHFYwWetc2khV5NogiPWlGHa4yujSZIQC3+j/zJ7TB6vuQcM2E+JevPHgYhhwmX1lZmixtIGEkX1c9A1ZFY=[/tex]（1）令[tex=8.786x2.786]gA9kcq16tsV+7Wjk8w05PpXF/yrseLDPjlhfODEvZn5PsdoZuckCvnOOPn6iyUQJYSl6KMcdpRuZYNyHgwPtKxrl9hrUUTVZsEFftdVWG8W7lt3YU91xIV8ALtuH6Q+0[/tex]。（2）带入（1）得[tex=12.071x2.786]075gCzZzsMRb6HYXYk9X90LZzs4JoSP/SYL9GheVTufiViyktMDMZMQEKEzSWI2zTB3a7z66YV3Kr5/UkCufyHsY4rPN0F+sm31CfeX5W1NNC4y5d5d/whyHHcPq5lunqkyrx5VCK8xnMwGtV5DCNDPJCE00oBbLwdHuOMi6Oy929k76XPpjg/i8i5UdKX6qsY81TAzB5jgDpoVeF/I1xQ==[/tex]（3）由（3）式得，[tex=5.857x1.429]OpdTdKo7cIYKYYSs8jFUQBWGTYoWGO+mo8PejCIrlQM=[/tex]。从而[tex=8.5x2.786]c9yWS5PljT+dkW/F5i0n5DD5t+lkyvCS+CPj9nipTi5qx7qlw20qe08ZiHxeKtTR/7b2uJA70waVJCn6BIuU5xxkjMpAq/LvxfLqXmOUG7OKb4hbCKdPWp0x44OZ9MGs[/tex]。于是[tex=23.214x3.357]GOgGvXf8fpWrP7XGIdsj8/VG/xNVh98rDAir4+82FjtqUNATgjIFIdt53aAM64qz06GhuDFTBFD9IXy+w1/3WGDOWocm66HxElLaWat2MwRtF26B4K4kMszHqv7S4/3AgceGHFoLEWrBm0zK80YeaaneLSRMzzfTEUWpVnnV0up955tVWIX/9eHVm9aT3+XZojPvC7pl2OxdLON++Cj9w+Et74VvmM1v4KTm0AZpYZ8wO9NEutyHyg4/jZv+FhuL0AETfMmythr+JkeMXLFJacQxRNgil0W4vaVOk56GOG7Qxbr2CLtemYe0fNyhBWMepujyPNebtzLV24eGEr7Jj3bJKUDk/YdvGs3l8x6htl4=[/tex]（4）因此[tex=23.714x2.786]lDSQTb/JbI7L/zeru0uz1uqVS/yXKTzi49m7HL9I4gjOuGYkBroQCUC01UGXKlaT+fYBFzYovenmBzFeEOOFMn8xDozGo0h29kajLDLqizLERz1IZajaKFjN5I9wsNThRX92ev3d2tPH40fmkq1SW1JuQ2En7lSrN51WU8CNsVm2whxNe3AbwIVgoTIOk+y9E2UVzW535VM22xluJfGGA1n+xi1pcQpQ4Nv2uYZ7DmiVz1ej64E4797js7txW+5++2liOisf/N+OibcURBCXvw==[/tex]。从而[tex=1.214x1.214]o+FNS96A6h2tHl0lOaYvdg==[/tex]是可逆的[tex=0.5x0.786]Tg0I1PUwmDJ7uXa9+yiYMA==[/tex]级对称矩阵。且[tex=38.143x3.143]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[/tex][tex=41.714x3.143]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[/tex][tex=36.071x3.143]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[/tex][tex=19.786x2.929]jaqkKt8zEwr29/cypwF+ryElrdkcvu/ELAZAJpGQKsvEGviWjEafXh6UrlGvYOq5OVpATfUDbFTZrQSScmpayYNA6eJLsZgrGMYLvd7NkHikthaSsciJkPwfE2Nwy+F5Cn601miapyykvTLs41YdkMHcWE6I+oifxgLm+jQe6YgwV+EZB4A/0O3C2lAfr5k2F3nx7Q9DXd/FvHQzLgtnQfbtxM/ArbTjl/0TLwiP8OFQubNo596hwXCQrYARQ+IQmk5Z+S8SRunCrZdiv6OpA+7IF2LTAlY42Tog/zBSuZo=[/tex][tex=12.571x3.071]oAHr35nPrigY+1IW2aWFbaR4uzeJNehA7hSlLUcgqqGEsOe9vSA0vnq3xBCOgxm+XUzNbtJAsQCJsan/TCO10qryc/TJtpPpKd9O8mODY6FxQe8p4IP+h5RRL+Y2p4kYbAV8aGleC98ELJXbSuZfKqfBYC6wGoYPePs0tXCctok4TNV7vtY5ciWj7ftULklJ[/tex]（5）由（5）式看出，[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的任一不等于0的[tex=0.5x0.786]Tg0I1PUwmDJ7uXa9+yiYMA==[/tex]阶主子式都与[tex=1.714x1.429]Ip0D0+pyHosF3muJRzoj7PILFQpBpgalD2ZFc3x0C6s=[/tex]同号。
举一反三
- 设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是实数域上的[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级对称矩阵，且[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的秩为[tex=3.0x1.357]jGI6hkgva7Rcyr50NnHREw==[/tex]。证明：[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]至少有一个[tex=0.5x0.786]Tg0I1PUwmDJ7uXa9+yiYMA==[/tex]阶主子式不为0.
- 设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是实数域上的[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级矩阵证明：如果[tex=3.429x1.357]KfxiXgR+wZCad+SOlQefBQ==[/tex]，且[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的每一个元素等于它自己的代数余子式乘以-1，那么[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是正交矩阵.
- 设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是实数域上的[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级矩阵。证明：如果[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的所有顺序主子式都大于0，且[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的所有非主对角元都小于0，那么[tex=1.714x1.214]d+9NDUvA5ZDrRGeFW5fxcQ==[/tex]的每个元素都大于0.
- 设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是实数域上的[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级矩阵证明：如果[tex=2.643x1.357]xnNlsIp2wAAq+OkAnU/oIQ==[/tex]，且[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的每一个元素等于它自己的代数余子式，那么[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是正交矩阵.
- 证明：如果[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级正交矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是上三角矩阵，那么[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是对角矩阵，且[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的主对角元为1或-1.
内容
- 0
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是复数域上的[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级矩阵，并且[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的元素全是实数。
- 1
设 [tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex] 为 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 阶实矩阵, 满足 [tex=3.643x1.214]u9ZFFjrmdLitRdLiKCtqhjog7ZeYbiv+qENyuyHI7/w=[/tex], 求证: [tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex] 是对称矩阵.
- 2
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是数域[tex=0.857x1.0]eMszuSG5by5UfRZVROYp5A==[/tex]上的[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级可逆矩阵，证明：如果[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]有特征值，那么[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的特征值不等于0.
- 3
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是一个[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级正交矩阵，证明：如果[tex=2.643x1.357]xnNlsIp2wAAq+OkAnU/oIQ==[/tex]，且[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]是奇数，那么1是[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的一个特征值。
- 4
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是一个[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级正交矩阵，证明：如果[tex=3.429x1.357]KfxiXgR+wZCad+SOlQefBQ==[/tex]，那么-1是[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]得一个特征值。