设n阶矩阵A的任意一行的元素之和都是a 证明a是矩阵A的一个特征值 求a对应的特征向量
\u5047\u5b9an\u9636\u77e9\u9635A\u7684\u4efb\u610f\u4e00\u884c\u4e2d\uff0cn\u4e2a\u5143\u7d20\u7684\u548c\u90fd\u662fa\uff0c\u8bd5\u8bc1\u03bb=a\u662fA\u7684\u7279\u5f81\u503c\uff0c\u4e14\uff081\uff0c1\uff0c\u2026\uff0c1\uff09T\u662f\u5bf9\u5e94\u4e8e\u03bb=a\u7684\u89e3\u7b54\uff1a\u8bc1\u660e\uff1a\u5047\u8bbe\uff1aA=a11\u2026a1n???an1\u2026ann\uff0c\u5219\uff1ank\uff1d1aik\uff1da\uff0c\uff08i=1\u2026n\uff09\uff0c\u4ee4\u5411\u91cf\uff1a\u03b1=\uff081\uff0c1\uff0c\u2026\uff0c1\uff09T\uff0c\u5219\uff1aA11?1=nk\uff1d1a1knk\uff1d1a2k?nk\uff1d1ank=aa?a=a11?1\uff0e\u4ece\u800c\uff1a\u03bb=a\u4e3aA\u7684\u7279\u5f81\u503c\uff0c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u4e3a\uff1a\uff081\uff0c1\uff0c\u2026\uff0c1\uff09T\uff0e\uff082\uff09\u56e0\u4e3aA\u53ef\u9006\uff0c\u6240\u4ee51a\u4e3aA-1\u7684\u7279\u5f81\u503c\uff0c\u4e14\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u4e5f\u662f\uff1a\uff081\uff0c1\uff0c\u2026\uff0c1\uff09T\uff0c\u5373\uff1aA?111?1\uff1d1a11?1\uff0c\u4e5f\u5c31\u662f\uff1ab11\u2026b1n???bn1\u2026bnn<table style="text-align: left; width: 100%; margin-left: 1px; margin-right: 1px" cellspacing="-1" ce
\u8003\u8651\u5217\u5411\u91cfx=(1, 1, ..., 1)
\u5b83\u548c\u8be5\u77e9\u9635\u7684\u4e58\u79ef\u662f(a,a,...,a)
\u5b83\u6ee1\u8db3Ax = ax\uff0c\u56e0\u6b64a\u662f\u7279\u5f81\u503c\uff0cx\u662f\u7279\u5f81\u5411\u91cf
它和该矩阵的乘积是(a,a,...,a)
它满足Ax = ax，因此a是特征值，x是特征向量
显然有A*[1 1 ...1]^T=[a a ...a]^T=a[1 1 ...1]^T,这里[1 1 ...1]^T是个n×1的向量。由此看出a是个特征值，而[1 1 ...1]^T是a所对应的特征向量。
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