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\(A\)为方阵,\(x_1, x_2\)分别为\(\lambda_1, \lambda_2\)对应的特征向量,\(\lambda_1 \neq \lambda_2\).
不同特征值对应的特征向量线性不相关, 即\(x_1, x_2\)线性不相关
假设\(x_1, x_2\)线性相关, 则存在非0值\(k\)使得\(x_1 = k x_2\)成立. 可得:
\[A x_1 = k A x_2 \]
\[\lambda_1 x_1 =k \lambda_1 x_2 = k \lambda_2 x_2\]
\[\lambda_1 = \lambda_2\]
矛盾.
对称阵不同特征值对应的特征向量不仅不线性相关, 还相互垂直, 即当\(A = A^T\)时,\(x_1 ^ Tx_2 = 0\)
\[ x_1^T x_2 = \frac {x_1 ^ T \lambda_2 x_2}{\lambda_2} \\= \frac {x_1 ^ T A x_2}{\lambda_2} \\= \frac {x_1 ^ T A^T x_2}{\lambda_2} \\= \frac {x_2^T A x_1 }{\lambda_2} \\= \frac {x_2^T \lambda_1 x_1 }{\lambda_2} \\= x_2^Tx_1 \frac {\lambda_1 }{\lambda_2} \]
要么\(\frac {\lambda_1 }{\lambda_2} = 1\), 要么\(x_1 ^ Tx_2 = 0\), 而\(\lambda_1 \neq \lambda_2\), 所以只能是\(x_1 ^ Tx_2 = 0\).
转载于:https://www.cnblogs.com/dengdan890730/p/6152812.html