(A)为方阵, (x_1, x_2)分别为(lambda_1, lambda_2)对应的特征向量, (lambda_1 eq 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 eq lambda_2), 所以只能是(x_1 ^ Tx_2 = 0).