1._本章公式汇总

标准正交化

将一线性无关的向量组 (以 α1,α2,α3\alpha_1, \alpha_2, \alpha_3 为例) 化为标准正交向量组的方法: 步骤 1. 施密特正交化:

β1=α1,β2=α2−(α2,β1)(β1,β1)β1,β3=α3−(α3,β1)(β1,β1)β1−(α3,β2)(β2,β2)β2.\begin{aligned} & \boldsymbol{\beta}_1=\boldsymbol{\alpha}_1, \\ & \boldsymbol{\beta}_2=\boldsymbol{\alpha}_2-\frac{\left(\boldsymbol{\alpha}_2, \boldsymbol{\beta}_1\right)}{\left(\boldsymbol{\beta}_1, \boldsymbol{\beta}_1\right)} \boldsymbol{\beta}_1, \\ & \boldsymbol{\beta}_3=\boldsymbol{\alpha}_3-\frac{\left(\boldsymbol{\alpha}_3, \boldsymbol{\beta}_1\right)}{\left(\boldsymbol{\beta}_1, \boldsymbol{\beta}_1\right)} \boldsymbol{\beta}_1-\frac{\left(\boldsymbol{\alpha}_3, \boldsymbol{\beta}_2\right)}{\left(\boldsymbol{\beta}_2, \boldsymbol{\beta}_2\right)} \boldsymbol{\beta}_2 . \end{aligned}

步骤 2. 规范化（单位化):

γ1=β1∥β1∥,γ2=β2∥β2∥,γ3=β3∥β3∥.\gamma_1=\frac{\boldsymbol{\beta}_1}{\left\|\boldsymbol{\beta}_1\right\|}, \gamma_2=\frac{\boldsymbol{\beta}_2}{\left\|\boldsymbol{\beta}_2\right\|}, \boldsymbol{\gamma}_3=\frac{\boldsymbol{\beta}_3}{\left\|\boldsymbol{\beta}_3\right\|} .
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