高等代数一道习题分享
作者：数疯菌题目：设A是n阶实矩阵, https://www.zhihu.com/equation?tex=A%3D%28a_%7Bij%7D%5E%7B%7D%29 ，且满足： https://www.zhihu.com/equation?tex=%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7Bij%7D%5E%7B%7D%7D%3D1%2Ci%3D1%2C2%2C...n%3B https://www.zhihu.com/equation?tex=a_%7Bij%7D%5E%7B%7D%5Cgeq+0%2Ci%2Cj%3D1%2C2%2C...n%3B求证：对于 https://www.zhihu.com/equation?tex=%5Clambda+%5Cin+R ,若存在n维列向量 https://www.zhihu.com/equation?tex=%5Cxi%5Cne0 使得https://www.zhihu.com/equation?tex=A%5Cxi%3D%5Clambda%5Cxi ，则 https://www.zhihu.com/equation?tex=%7C%5Clambda%7C%5Cleq1 。(引：其实满足这样条件的矩阵A的转置 https://www.zhihu.com/equation?tex=A%5E%7BT%7D 为马尔可夫矩阵，它有性质为：（1） https://www.zhihu.com/equation?tex=%5Clambda%3D1 是其最大的特征值。（2）其余特征值的绝对值小于1.（3） https://www.zhihu.com/equation?tex=%EF%BC%88A%5E%7BT%7D%29+%5E%7Bk%7D 也是马尔可夫矩阵，且每个元素非负，每列元素和为1，读者可以查找马尔可夫矩阵英文原文献）证明：首先， https://www.zhihu.com/equation?tex=%EF%BD%9C%5Clambda+E-A%EF%BD%9C%3D%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D+%5Clambda-a_%7B11%7D%26a_%7B12%7D+...%26a_%7B1n%7D%5C%5Ca_%7B21%7D%26%5Clambda-a_%7B22%7D...%26a_%7B2n%7D%5C%5C...%26...%26...%5C%5Ca_%7Bn1%7D%26a_%7Bn2%7D...%26%5Clambda-a_%7Bnn%7D%5Cend%7Barray%7D+%5Cright%7C ,将第2，3，...n列全部加到第一列，得到：
https://www.zhihu.com/equation?tex=%EF%BD%9C%5Clambda+E-A%EF%BD%9C%3D%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D+%5C+%5Clambda-1%26a_%7B12%7D+...%26a_%7B1n%7D%5C%5C%5Clambda-1%26%5Clambda-a_%7B22%7D...%26a_%7B2n%7D%5C%5C...%26...%26...%5C%5C%5Clambda-1%26a_%7Bn2%7D...%26%5Clambda-a_%7Bnn%7D%5Cend%7Barray%7D+%5Cright%7C ，第一列提出（ https://www.zhihu.com/equation?tex=%5Clambda-1 ），则有：https://www.zhihu.com/equation?tex=%EF%BC%88%5Clambda-1%EF%BC%89 https://www.zhihu.com/equation?tex=%EF%BD%9C%5Clambda+E-A%EF%BD%9C%3D%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D+%5C+%5C+1%26a_%7B12%7D+...%26a_%7B1n%7D%5C%5C1%26%5Clambda-a_%7B22%7D...%26a_%7B2n%7D%5C%5C...%26...%26...%5C%5C1%26a_%7Bn2%7D...%26%5Clambda-a_%7Bnn%7D%5Cend%7Barray%7D+%5Cright%7C =0，所以A必有特征根 https://www.zhihu.com/equation?tex=%5Clambda%3D1 ，
令A= https://www.zhihu.com/equation?tex=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D+%5C+a_%7B11%7D%26a_%7B12%7D+...%26a_%7B1n%7D%5C%5Ca_%7B21%7D%26%5C+a_%7B22%7D...%26a_%7B2n%7D%5C%5C...%26...%26...%5C%5Ca_%7Bn1%7D%26a_%7Bn2%7D...%26%5C+a_%7Bnn%7D%5Cend%7Barray%7D+%5Cright%5D ，则 https://www.zhihu.com/equation?tex=A%5E%7B2%7D%3DA%5Ctimes+A%3D https://www.zhihu.com/equation?tex=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D+%5C+a_%7B11%7D%26a_%7B12%7D+...%26a_%7B1n%7D%5C%5Ca_%7B21%7D%26%5C+a_%7B22%7D...%26a_%7B2n%7D%5C%5C...%26...%26...%5C%5Ca_%7Bn1%7D%26a_%7Bn2%7D...%26%5C+a_%7Bnn%7D%5Cend%7Barray%7D+%5Cright%5D https://www.zhihu.com/equation?tex=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D+%5C+a_%7B11%7D%26a_%7B12%7D+...%26a_%7B1n%7D%5C%5Ca_%7B21%7D%26%5C+a_%7B22%7D...%26a_%7B2n%7D%5C%5C...%26...%26...%5C%5Ca_%7Bn1%7D%26a_%7Bn2%7D...%26%5C+a_%7Bnn%7D%5Cend%7Barray%7D+%5Cright%5D =https://www.zhihu.com/equation?tex=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D+%5C+%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7B1j%7Da_%7Bj1%7D%7D%26%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7B1j%7Da_%7Bj2%7D%7D...%26%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7B1j%7Da_%7Bjn%7D%7D+%5C%5C+%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7B2j%7Da_%7Bj1%7D%7D%26%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7B2j%7Da_%7Bj2%7D%7D...%26%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7B2j%7Da_%7Bjn%7D%7D%5C%5C...%26...%26...%5C%5C%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7Bnj%7Da_%7Bj1%7D%7D%26%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7Bnj%7Dx_%7Bj2%7D%7D...%26%5C+%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7Bnj%7Da_%7Bjn%7D%7D%5Cend%7Barray%7D+%5Cright%5D ,则每行元素相加，有：https://www.zhihu.com/equation?tex=%5Csum_%7Bk%3D1%7D%5E%7Bn%7D%7B%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7Bij%7Da_%7Bjk%7D%7D%7D%3D%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7Bij%7D%7D%5Csum_%7Bk%3D1%7D%5E%7Bn%7D%7Ba_%7Bjk%7D%7D%3D1 , https://www.zhihu.com/equation?tex=i%3D1%2C2...n%3B 所以 https://www.zhihu.com/equation?tex=A%5E%7B2%7D 每行元素和为1，由数学归纳法，易得： https://www.zhihu.com/equation?tex=A%5E%7Bk%7D 的每行元素和为1.又因为 https://www.zhihu.com/equation?tex=A%5Cxi%3D%5Clambda%5Cxi ，则有 https://www.zhihu.com/equation?tex=A%5E%7Bk%7D%5Cxi%3D%5Clambda%5E%7Bk%7D%5Cxi .假如 https://www.zhihu.com/equation?tex=%5Clambda%3E11" style="max-width: 100%; vertical-align: middle; margin-right: 3px; margin-left: 3px; display: inline-block;"> 成立,当k充分大时，上述等式右边向量中元素趋于无穷大，但左边不会趋于无穷大，所以， https://www.zhihu.com/equation?tex=%EF%BD%9C%5Clambda%EF%BD%9C%3E11" style="max-width: 100%; vertical-align: middle; margin-right: 3px; margin-left: 3px; display: inline-block;"> 时，等式 https://www.zhihu.com/equation?tex=A%5E%7Bk%7D%5Cxi%3D%5Clambda%5E%7Bk%7D%5Cxi 不成立，所以假设错误，故而 https://www.zhihu.com/equation?tex=%EF%BD%9C%5Clambda%EF%BD%9C%5Cleq1 。由上题，可以得到推论：若一个矩阵每行和为一个定值K，则这个矩阵必有一个特征值为K。数疯菌对这个题作出改编：（原创）改编题目：设A是n阶实矩阵, https://www.zhihu.com/equation?tex=A%3D%28a_%7Bij%7D%5E%7B%7D%29 ，且满足： https://www.zhihu.com/equation?tex=%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%EF%BD%9C%7Ba_%7Bij%7D%EF%BD%9C%5E%7B%7D%7D%3D1%2Ci%3D1%2C2%2C...n%3B https://www.zhihu.com/equation?tex=a_%7Bij%7D%5E%7B%7D%5Cgeq+0%2Ci%2Cj%3D1%2C2%2C...n%3B求证：对于 https://www.zhihu.com/equation?tex=%5Clambda+%5Cin+R ,若存在n维列向量 https://www.zhihu.com/equation?tex=%5Cxi%5Cne0 使得https://www.zhihu.com/equation?tex=A%5Cxi%3D%5Clambda%5Cxi ，则 https://www.zhihu.com/equation?tex=%7C%5Clambda%7C%5Cleq1 。证明：令A= https://www.zhihu.com/equation?tex=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D+%5C+a_%7B11%7D%26a_%7B12%7D+...%26a_%7B1n%7D%5C%5Ca_%7B21%7D%26%5C+a_%7B22%7D...%26a_%7B2n%7D%5C%5C...%26...%26...%5C%5Ca_%7Bn1%7D%26a_%7Bn2%7D...%26%5C+a_%7Bnn%7D%5Cend%7Barray%7D+%5Cright%5D https://www.zhihu.com/equation?tex=%5Cxi%3D%5Cleft%5B+b_%7B1%7D%5C%5C+b_%7B2%7D+%5C%5C.%5C%5C.%5C%5C.%5C%5Cb_%7Bn%7D%5Cright%5D即： https://www.zhihu.com/equation?tex=A%3D%28a_%7Bij%7D%5E%7B%7D%29 , i,j=1,2,...n;https://www.zhihu.com/equation?tex=%5Cxi%3D%28b_%7Bq%7D%29 ,q=1,2,...n;则 https://www.zhihu.com/equation?tex=A%5Cxi%3D%5Clambda%5Cxi 推出：https://www.zhihu.com/equation?tex=%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7Bqj%7D%7Db_%7Bj%7D%3D%5Clambda+b_%7Bq%7D%2Cq%3D1%2C2%2C...n%3B 则有： https://www.zhihu.com/equation?tex=%EF%BD%9C%5Clambda+b_%7Bq%7D%EF%BD%9C%3D+%EF%BD%9C%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7Bqj%7D%7Db_%7Bj%7D%EF%BD%9C%5Cleq+%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%EF%BD%9C%7Ba_%7Bqj%7D%7Db_%7Bj%7D%EF%BD%9C%3D%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%EF%BD%9C%7Ba_%7Bqj%7D%EF%BD%9C%EF%BD%9C%7Db_%7Bj%7D%EF%BD%9C ，i=1，2，...n;设M=max{ https://www.zhihu.com/equation?tex=%5Cleft%7C+b_%7Bq%7D+%5Cright%7C },则推出： https://www.zhihu.com/equation?tex=%EF%BD%9C%5Clambda%EF%BD%9C%EF%BD%9Cb_%7Bq%7D%7C%5Cleq+%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7C%7Ba_%7Bij%7D%7D%7C%7CM%7C ,即有：https://www.zhihu.com/equation?tex=%EF%BD%9C%5Clambda%EF%BD%9C%5Cleq+%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7C%7Ba_%7Bij%7D%7D%7C%5Cfrac%7B%EF%BD%9CM%EF%BD%9C%7D%7B%EF%BD%9Cb_%7Bq%7D%7C%7D ，因为要对任意的 https://www.zhihu.com/equation?tex=%5Cleft%7C+b_%7Bq%7D+%5Cright%7C 都成立，所以 https://www.zhihu.com/equation?tex=%EF%BD%9C%5Clambda%EF%BD%9C%5Cleq+%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7C%7Ba_%7Bij%7D%7D%7C%5Ctimes1%3D1 ,i=1,2,...n;欢迎大家关注数疯菌知乎号和微信公众号：数院经典题，数疯菌将在这两个平台分享数学总结和好题分析。
66666666666666666
页:
[1]