[tex=1.857x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 皆为 [tex=2.429x1.071]/1altEfb7YJTb68/W6737A==[/tex] 复矩阵,证明:方程 [tex=4.0x1.0]iBuDKI65/og8QoaCBkLEgw==[/tex] 有非零解的 充分必要条件是[tex=2.0x1.214]Fu8IbDhz8zi1Fz+2a1sLzg==[/tex]有公共特征值.
矩阵方程 [tex=4.0x1.0]OxxakEMUXnNUbkgw1KPV9g==[/tex]可以写成齐次线性方程组.设 A=[tex=14.643x1.5]E3huwMzQ5+ygoDpWoboHIKJNiFM9njiWoiJN3lXjW0nrtAoiGGlvFETGEmiyT2nEb93OQ+LPPUWE9Ss9/LyCf0XXt5Eod+2WgRcBgUQHIGsIl84QIjb9+oLt56KWYDQSgQLRETC+/c6DtwBUPhdSgzF39GzE4PzsH9QOehW+UpIO20GubQt9VW3Dh7B0hT95[/tex] 把 [tex=0.857x1.0]N7iCrOsS+NNEUUlnsYCi1g==[/tex] 对应到下面的 [tex=2.714x1.357]k6g5iSqAXCQhTVmrg/t7yA==[/tex] 向量 [tex=8.786x12.357]Bnq69O8ZCnTnc5QS4LG5XQ4idFppfQyYYbB/9wRJq2egeeELCUq4Ea2NIQ3hfzFRB43z5DaERFZh8eY02g+TzpgEG3m0BwiozPx7sZIlddgQ8gYHioND6s7BIV3cFQbGUX8NKkEhlzaYfhPrdj2wZJYJj0iFVYvTWzLY8JVYF30lRceifRdmQnBPrcK9EuzyqiUw72I4+qWJybU5pg7OHUckjuYOx/IBHVheFdPrW1QgG6Vs5x3IBvGBqXIZshCK[/tex]经过计算,上面的矩阵方程等价于下面的方程组 [tex=11.286x4.786]THLoc4JiHqtEuDF0smAtg6MbtkFrK5YF98asw0ZbZbTz2iAOlIWYt25Wb7v+aCjB8Ki0zrx/1JyHq2iicYySPKQS3HRIDnr4YzpImkFOWd83BlQ5/fAXA1+FyPFwTIgJkQvQQ4PC36TYcTE4WE2s2XvXVpEDfUFo8cZU4WA4wT1BJS60rueFkFRZ8Iab07qffxe/ViFWGUxE797Bt5dtZQ==[/tex][tex=18.571x5.357]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[/tex] 又 [tex=4.0x1.0]OxxakEMUXnNUbkgw1KPV9g==[/tex] 与[tex=17.286x1.571]7ksc2ZNjj/vmGJUWhYyVhP2tgpc3zaQJh35dx9XbY4ZTo9FnL7XXx/IJZ9Slx36+5qcZqvCvxx5ClajPjF1Xzdu++U8n2pXpE19IG8qA2dlEn4ox6fc6jjeuCjdWNZiZ[/tex] 同时有非 零解. 对复矩阵 [tex=0.929x1.0]ep004cu6Ev4qhlMpamsNGg==[/tex]有可逆的 [tex=0.643x1.0]7p9LSMKGuAkm+tarwSUSAw==[/tex]使 [tex=3.071x1.214]jNJUVyn923+eHJN4sq2BY5xuJqjdUUE31UEZ9ErPkHE5GxpvUo6vdqn9rAS9O7Lq[/tex]成上三角形 [tex=10.857x5.929]075gCzZzsMRb6HYXYk9X97UCDFKVjkRxpfyvgaqvfb1oCA5M+64F77zq/lyxUodxAXnu6MjhkF5t0KN0SMCMX0ke/UcFwJBXU/vgQyJBlldQRCkeEVA2WuBivCmupEvspdbkFaLp2fJM/+x8wdLk9w2Uxhy7gOzgJcO0V30kN8fuK/JOgeIUJ1eim/58z72N[/tex]我们记[tex=23.214x1.643]YVl2sylvTLr7dbv5ZICe3SUQ0vJpQ5r+LHgAbjq+odMrUJhyw+qIUybU45KV8HKdZX89yw9UEy+obQmNxQgFUPEPSrH6VIqOyjgXyJdCV01eXiZxVIlvObDBULgGgGFttMLJ0K2qYgUajB9ys5K7/g==[/tex][tex=3.714x1.643]sJxzLtg24e0HClxHSlZ+2ekYQhQSc8GsNBhkBmmArno8SkrE5pdnovK8Ly64UnKc[/tex] 则[tex=5.357x1.214]puSIJGPzubTQt3a5mpJL+etIMUG6WvbRan67QIoXbSs=[/tex]与下面的齐次线性方程组等价[tex=12.357x5.214]THLoc4JiHqtEuDF0smAtg6MbtkFrK5YF98asw0ZbZbT8777YW3WBJ8MDooWCae36m7V/z038DQTldXtKhYVb9RBm9izm9dH0YmTKek92n5mm/Wn9+vMebofBF4ancQ83X5JEW/jjpmEFE9SJ0OdSEJI+vxgsW+nBupPbW/nCZHXIMv/MNbCD4bN1SW727WHFjF5ORq4HzbdGcSOhgrr99SsYw+mAUURRWWxjJbX3jNM=[/tex][tex=19.214x5.929]yZSoi3OqRDHA0jMpBBfkrVBoAk7hKFAtm8DJc/NCrxjDhNWaFuKmR2HySmg1eZpnN1UZodIR8+XCTv/TH7N/3laeql/UfkMmLn7Kvil47V6RgVQdBVmeACvtP1FJKiaOc57vwR0hh0ltcVSKkY5oUjiZlXhNmcVyM2WKl8thrnuLNSMtSlw5Os8QZDJoRest5IFvSgmFflWkUI+LkGLkCHM36EdudmzYmwfhCS7RoYJBSP4erjWgUXTnBNEp6hncrD/vwFKIOK7+5c1FaoLf+d2bsY2H12Lcd84zNgBlKerkbQ4p/PlRznJS1mf1aNiIEjEnlC9vK9IjicrDHOJYi4audOoRjNLy9+uEM7mkUIFvU1T7SxqrsljF5qCoYCZYyL3AJkgsnJ8YVA+3t+O6gTa2pugci8vTkAZMX6pOaY/oUO1bekNrIfTMBjvDWJqyF3w26autJddf59Hxg9rEtLLwbmKrXoUGCpdnJ2g8fucAjx3vr3iR1RcQlbUJ4bOyOuXAwV6JsED0p2Mmkbt7XRf/5g5wDjr8+WNltyg5knBTHfbTjGJqOOnxslk14QtGMPTLiyJw/4p+QkO3keQvYc93FTAJ1anzB/OSNXp8Tlk=[/tex]即[tex=24.0x5.786]075gCzZzsMRb6HYXYk9X9+IcXtUt2GhKaIWnmuGg9s8c44pVqEffBiiSQgxdK7DnisF9ddlW7Gz8PxMBm+Zn9Fa8m1l1Vli3qPW8D38BZhT5YOV1hUeAJLIRlDBEZ7zUpT1JvKUAJY3UQXJNpm2n+SbyjTxkNHNijGtKoARJ/tH4jkC671BAcFTvvIlKbyPlc+D1Oiv7+59T5hEKOELso9KK7XstQBHXB+jfqzBWRqqUX9JLbTP9Kl3YYyFNjdAioIjqM3h3cmtV84vU+UdPPLUvH0TI/BgOJBvgR0M36ptf9pWM6VHE/+CFxvv7RtFFOuk+tdbYbyVfZEKnJAaPONbzKOTqw0rWuD2oKpQqZcKVruSukup4E9PJuzrbiWct[/tex]方程有非零解的充分必要条件是系数行列式为零,即 [tex=7.286x3.286]AjAO8MG/VMMNdeaBs5QvAMmrEahWpGh4eE2fhW4bIgdzQhu2+A28G9SMKyVn1jSg9Xykbzsrn4EEAZAvaqYD9YtrqVEkwV6W7pdxa7dm2Bw=[/tex]也即有某 [tex=0.929x1.357]vU+jC7MuG2jtBqobxwNhQA==[/tex] 是 [tex=1.143x1.214]Z3brU/WMYTtkfYvo+Q4+fw==[/tex]的特征值. 由 [tex=0.929x1.0]ep004cu6Ev4qhlMpamsNGg==[/tex] , 是上三角形矩阵,与[tex=0.929x1.0]Wc7F6pzA+24VkPyxde7oQQ==[/tex]相似,故 [tex=1.071x1.5]wWg/3Fean5XapiV2IUOlAbgw+kEubzA780k2GEtPF6M=[/tex] 是[tex=0.786x1.0]sHo1pKm+gjxjcUAJjHrarQ==[/tex] 的一个特征值,又 [tex=1.143x1.214]R0Bx+ybSEpubLRkiLymmmA==[/tex] 与 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 相似[tex=1.429x1.214]wMjwWv7wmzS5xoYz4i2h2Q==[/tex] 的特征值是 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 的特征 值. 这说明条件 [tex=1.286x1.357]VXS75MzTkkVQBNaBAz9l9g==[/tex]是 [tex=0.786x1.0]sHo1pKm+gjxjcUAJjHrarQ==[/tex]与[tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex]有公共特征值.又[tex=1.286x1.357]utHvH4igaptAEXsEZc1Kjw==[/tex]与[tex=1.286x1.357]GEp98LAjkJwaCSrpbBogIw==[/tex]的等价性说明它们同时有非零解. 且[tex=1.286x1.357]utHvH4igaptAEXsEZc1Kjw==[/tex] 与[tex=4.0x1.0]NOPcm8UWnZHj7FrEql91JA==[/tex] 等价,说明[tex=1.286x1.357]utHvH4igaptAEXsEZc1Kjw==[/tex]有非零解即[tex=4.0x1.0]9uTdcDwOQd4uorxSQysCcw==[/tex] 有非零解. 最终得到[tex=4.0x1.0]NOPcm8UWnZHj7FrEql91JA==[/tex]有非零解的充分必要条件是 [tex=2.0x1.214]vnzjVhyzo/NIhVUgFyjLlA==[/tex]有公共的特征值.
举一反三
- 设 [tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 均为[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 阶方阵,证明 [tex=1.786x1.0]96v/QovsZC+SUffULwqQnvkECUDsFAQTkNLNhDaRrpw=[/tex] 与 [tex=1.786x1.0]kvRlRS1CLNU1fC2siJ4VO2nX6gBe0vc1kFMLW9VYZjM=[/tex] 有相同的特征值.
- 设 [tex=7.143x2.786]g+/KVfaQxdXa8hxnv147pHWU52BSD9LdtB9aAf9v+Sbw0blXLyIcHBU0jSX5ERGgSQrByRePHgOQ3Dstda1AfQKRafS7BreZoZMXKC8Vbf2UFrXzBeXPcDyK1mfGT//NllwrWZYMcqn7M6gzUNt+dxT4qlQcWezHyfAwkusJil4=[/tex] 其中 [tex=2.0x1.214]p/fPb4cKwKYaAJ8NhtZPtw==[/tex]分别是 [tex=1.929x1.0]FLsL1n4WDTNpV4dn6Kq2dg==[/tex] 阶矩阵 求证:若[tex=0.929x1.0]dS4Ce7aCn5Z2jKx4QASmCg==[/tex] 是正定矩阵[tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 都是正定矩阵,反之也成立.
- 证明: 设 [tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 都是 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 阶正交方阵,则[tex=2.714x1.357]RRZ9zlAN4pWdGS7d9wHOksQIkUN+jWMfk9arM96kBXA=[/tex] 或 [tex=1.286x1.143]Mj6+lbt3rBoas+xQLVX/oA==[/tex]
- 6个顶点11条边的所有非同构的连通的简单非平面图有[tex=2.143x2.429]iP+B62/T05A6ZTM0eeaWiQ==[/tex]个，其中有[tex=2.143x2.429]ndZSw3zT0QTOVLVdoUto1Q==[/tex]个含子图[tex=1.786x1.286]J+vVZa2YaMpc6mJBbqVvWw==[/tex]，有[tex=2.143x2.429]lmhx48evnQMhi03NovPXig==[/tex]个含与[tex=1.214x1.214]kFXZ1uR8GjycbJx+Ts2kyQ==[/tex]同胚的子图。供选择的答案[tex=3.071x1.214]3KinXFh3SXhZ7nIe1y9KEV6aadxhhJWeEy6Dij1iObdMUZkY6ZA5J2dVVjPSuhEf[/tex]：(1) 1 ;(2) 2 ;(3) 3 ; (4) 4 ;(5) 5 ;(6) 6 ; (7) 7 ; (8) 8 。
- 设 [tex=2.0x1.214]vnzjVhyzo/NIhVUgFyjLlA==[/tex] 分别是 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 级、[tex=0.929x0.786]D9maNLyVVGrC3QbL9jjRWg==[/tex] 级复矩阵, 证明: 矩阵方程 [tex=5.286x1.143]mDJ9zMFjT0VQSlihRefGTw==[/tex] 只有零解的充分必要条件是, [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 与 [tex=0.786x1.0]sHo1pKm+gjxjcUAJjHrarQ==[/tex] 没有公共的特征值.
内容
- 0
直接证明：若[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]与[tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]有公共的特征值，则矩阵方程[tex=4.0x1.0]wpEZ6UnL2rVoQItPDwH4Xw==[/tex]有非零解.
- 1
设[tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 是两个[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]阶实对称矩阵证明 [tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]与 [tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex] 相似的充要条件是 [tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]与 [tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex] 有相同的特 征值.
- 2
证明: 设[tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex]都是 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]阶正交方阵,则 [tex=5.571x1.429]317mMb/UfJBjZHDU7raSnlGWNv3TAfOvDYKp6rxGdYH/wkTdLKG3lnIOSFYz8youND3JkA/f56Zt6vj//KbUBaNtSDrFJ/TojTxEfphc2zw=[/tex] 也是正交方阵
- 3
已知[tex=5.0x1.286]nNRgYScRPw16N2lBJqtTsA==[/tex]，[tex=5.0x1.286]ZIJz5gTGIgdeWAGMFdoL1A==[/tex]，则[tex=6.214x1.286]wE5wtWoL9HR6uGPZrIzvHA==[/tex]成立的[tex=0.571x1.286]XubEW9+1+hkJqH7jXe5MrA==[/tex]值为 A: 1 B: 2 C: 4 D: 6 E: 8
- 4
有容量分别为[tex=3.286x1.286]pCZ+fPe3X5XtlIcXCf6RGw==[/tex]和[tex=3.286x1.286]JjWMjbwalVPPThZBywJsLQ==[/tex]的独立随机样本得到下述观测结果， （X、 Y为观测值， f为频数）X 12.3 12.5 12.8 13.0 13.5 Y 12.2 12.3 13.0f 1 2 4 2 1 f 6 8 2现已知变量X、Y的总体均呈正态分布。请问在0.05的显著性水平下，可否认为这两个总体属同一分布？[tex=24.786x1.286]OVWwFMgiPzBDnRSqBYypUv4puOxaqZVbzeGoYhEt/ZwiQxP0kGgAAWuaJInyBhH09xLkSWqB6n3qd1WXaKpfvwUNfmmVSMJTzi4wz4IT6q4=[/tex][tex=8.429x1.286]AcUD6cTXhAghaQMem3GRbFMfFVpZHcyA3tP0z+S7RAk=[/tex] [tex=13.357x1.357]ZPe8nXNlBeMmW2cEA+D6DaqP/loFbcVH2QukDH1SMofLM6E74nDyl0WrH8imm/Ai[/tex]