Gram-Schmidt 过程

• 正交化

  \(\beta_1=\alpha_1\)

  \(\beta_2=\alpha_2-\frac{(\alpha_2,~\beta_1)}{||\beta_1||}\frac{\beta_1}{||\beta_1||}=\alpha_2-\frac{(\alpha_2,~\beta_1)}{||\beta_1||^2}\beta_1=\alpha_2-\frac{(\alpha_2,~\beta_1)}{(\beta_1,~\beta_1)}\beta_1\)

  \(\beta_3=\alpha_3-\frac{(\alpha_3,~\beta_1)}{||\beta_1||}\frac{\beta_1}{||\beta_1||}-\frac{(\alpha_3,~\beta_2)}{||\beta_2||}\frac{\beta_2}{||\beta_2||}\\=\alpha_2-\frac{(\alpha_2,~\beta_1)}{||\beta_1||^2}\beta_1-\frac{(\alpha_2,~\beta_2)}{||\beta_2||^2}\beta_2\\=\alpha_2-\frac{(\alpha_2,~\beta_1)}{(\beta_1,~\beta_1)}\beta_1-\frac{(\alpha_2,~\beta_2)}{(\beta_2,~\beta_2)}\beta_2\)

  \(\cdots\)

  \(\beta_n=\alpha_n-\frac{(\alpha_n,~\beta_1)}{(\beta_1,~\beta_1)}\beta_1-\frac{(\alpha_n,~\beta_2)}{(\beta_2,~\beta_2)}\beta_2-\cdots-\frac{(\alpha_n,~\beta_{n-1})}{(\beta_{n-1},~\beta_{n-1})}\beta_{n-1}\)

• 单位化

  \(e_1=\frac{\beta_1}{||\beta_1||}\)

  \(e_2=\frac{\beta_2}{||\beta_2||}\)

  \(\cdots\)

  \(e_n=\frac{\beta_n}{||\beta_n||}\)