不同特征值对应的特征向量

\(A\)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\(x_1 ^ Tx_2 = 0\).