# 特征值和特征向量之间的关系

$$$$V_{\lambda_i} = \{\alpha \big| T(\alpha) = \lambda_i\alpha, \alpha \in V\}$$$$

$$$$V_{\lambda_i} = \{x \big| Ax = \lambda_ix, x \in C^n \}$$$$

$$V_{\lambda_i}$$$$C^{n\times n}$$的一个子空间，称$$V_{\lambda_i}$$是矩阵$$A$$1的对应于$$\lambda_i$$特征子空间

1. 矩阵$$A$$的任一特征值的几何重数不大于它的代数重数
2. 如果$$\lambda\_0$$的代数重数是1，则它的几何重数$$dim(V_{\lambda\_0})=1$$

# 不变子空间

1. 也即线性变换$$T$$

2. 根据特征向量的定义:$$(\lambda_iI-A)x_i=0$$

3. 因为$$Ax_j = \lambda_jx\_j$$,所以$$(\lambda_iI-A)x_j=(\lambda_i-\lambda_j)x_j$$

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