# 特征值

实对称矩阵的特征多项式的每一个复根都是实数，从而它们都是特征值。

证明：

$$A \boldsymbol{\alpha} = \lambda \boldsymbol{\alpha} \Rightarrow \left\{ \begin{aligned} & \boldsymbol{\alpha}^\top A \bar{\boldsymbol{\alpha}} = \boldsymbol{\alpha}^\top \lambda \bar{\boldsymbol{\alpha}} \\ & \boldsymbol{\alpha}^\top A \boldsymbol{\alpha}= \boldsymbol{\alpha}^\top \bar{\lambda} \bar{\boldsymbol{\alpha}} \end{aligned} \right . \Rightarrow \bar\lambda = \lambda$$