\(A\)为方阵, \(x_1, x_2\)分别为\(\lambda_1, \lambda_2\)对应的特征向量, \(\lambda_1 \neq \lambda_2\).
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.
\(x_1 ^ Tx_2 = 0\)
\[x_1^T x_2 = \frac {x_1 ^ T \lambda_2 x_2}{\lambda_2}
\\= \frac {x_1 ^ T A x_2}{\lambda_2}
\\= \frac {x_1 ^ T A^T x_2}{\lambda_2}
\\= \frac {x_2^T A x_1 }{\lambda_2}
\\= \frac {x_2^T \lambda_1 x_1 }{\lambda_2}
\\= x_2^Tx_1 \frac {\lambda_1 }{\lambda_2}
\]
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\(x_1 ^ Tx_2 = 0\).