标准正交基

定义

$$\boldsymbol \alpha = a_1 \boldsymbol \varepsilon_1 + a_2 \boldsymbol \varepsilon_2 + \cdots + a_n \boldsymbol \varepsilon_n = (\boldsymbol \varepsilon_1, \boldsymbol \varepsilon_2, \cdots, \boldsymbol \varepsilon_n) \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix}$$

$$\boldsymbol \beta = b_1 \boldsymbol \varepsilon_1 + b_2 \boldsymbol \varepsilon_2 + \cdots + b_n \boldsymbol \varepsilon_n = (\boldsymbol \varepsilon_1, \boldsymbol \varepsilon_2, \cdots, \boldsymbol \varepsilon_n) \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix}$$

$$(\boldsymbol \alpha, \boldsymbol \beta) = \sum_{i=1}^n \sum_{j=1}^n a_i b_j (\boldsymbol \varepsilon_i, \boldsymbol \varepsilon_j) = \\ (a_1, a_2, \cdots, a_n) \mathbf A \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix}$$

其中

$$\mathbf A = \begin{pmatrix} (\boldsymbol \varepsilon_1, \boldsymbol \varepsilon_1) & (\boldsymbol \varepsilon_1, \boldsymbol \varepsilon_2) & \cdots & (\boldsymbol \varepsilon_1, \boldsymbol \varepsilon_n) \\ (\boldsymbol \varepsilon_2, \boldsymbol \varepsilon_1) & (\boldsymbol \varepsilon_2, \boldsymbol \varepsilon_2) & \cdots & (\boldsymbol \varepsilon_2, \boldsymbol \varepsilon_n) \\ \vdots & \vdots & \ddots & \vdots \\ (\boldsymbol \varepsilon_n, \boldsymbol \varepsilon_1) & (\boldsymbol \varepsilon_n, \boldsymbol \varepsilon_2) & \cdots & (\boldsymbol \varepsilon_n, \boldsymbol \varepsilon_n) \\ \end{pmatrix}$$

称作基 $\boldsymbol \varepsilon_1, \boldsymbol \varepsilon_2, \cdots, \boldsymbol \varepsilon_n$ 的度量矩阵

若 $\mathbf A$ 为对角矩阵, 即 $(\boldsymbol \varepsilon_i, \boldsymbol \varepsilon_j) = \delta_{ij} = \begin{cases} 1, & i=j \\ 0, & i \neq j \end{cases}$ 向量两两正交, 则称该基为正交基

若 $\mathbf A$ 为单位矩阵, 即向量两两正交且每个向量都为单位向量, 则称该基为标准正交基

在标准正交基下,

$$(\boldsymbol \alpha, \boldsymbol \beta) = \boldsymbol \alpha^T \boldsymbol \beta$$

施密特正交法

若 $\boldsymbol \alpha_1, \boldsymbol \alpha_2, \cdots, \boldsymbol \alpha_s$ 是线性无关的向量组

1. 正交化

$$\boldsymbol \beta_1 = \boldsymbol \alpha_1$$

$$\boldsymbol \beta_2 = \boldsymbol \alpha_2 - \dfrac{(\boldsymbol \alpha_2, \boldsymbol \beta_1)}{(\boldsymbol \beta_1, \boldsymbol \beta_1)} \boldsymbol \beta_1$$

$$\boldsymbol \beta_3 = \boldsymbol \alpha_3 - \dfrac{(\boldsymbol \alpha_3, \boldsymbol \beta_1)}{(\boldsymbol \beta_1, \boldsymbol \beta_1)} \boldsymbol \beta_1 - \dfrac{(\boldsymbol \alpha_3, \boldsymbol \beta_2)}{(\boldsymbol \beta_2, \boldsymbol \beta_2)} \boldsymbol \beta_2$$

$$\vdots$$

$$\boldsymbol \beta_s = \boldsymbol \alpha_s - \dfrac{(\boldsymbol \alpha_s, \boldsymbol \beta_1)}{(\boldsymbol \beta_1, \boldsymbol \beta_1)} \boldsymbol \beta_1 - \cdots - \dfrac{(\boldsymbol \alpha_s, \boldsymbol \beta_{s-1})}{(\boldsymbol \beta_{s-1}, \boldsymbol \beta_{s-1})} \boldsymbol \beta_{s-1}$$

2. 单位化

$$\boldsymbol \eta_i = \dfrac{\boldsymbol \beta_i}{|\boldsymbol \beta_i|}$$

则 $\boldsymbol \eta_1, \boldsymbol \eta_2, \cdots, \boldsymbol \eta_s$ 为标准正交基