高等代数教学辅导

子节 8.2.1 主要知识点

1. 先把线性无关的向量组\(\alpha_1,\alpha_2,\cdots ,\alpha_s\)化成正交向量组\(\beta_1,\beta_2,\cdots ,\beta_s\)，令

   \begin{equation*} \begin{array}{ccl} \beta_1&=&\alpha_1,\\ \beta_2&=&\alpha_2-\frac{\left(\alpha_2,\beta_1\right)}{\left(\beta_1,\beta_1\right)}\beta_1,\\ \beta_3&=&\alpha_3-\frac{\left(\alpha_3,\beta_1\right)}{\left(\beta_1,\beta_1\right)}\beta_1-\frac{\left(\alpha_3,\beta_2\right)}{\left(\beta_2,\beta_2\right)}\beta_2,\\ &\vdots&\\ \beta_s&=&\alpha_s-\frac{\left(\alpha_s,\beta_1\right)}{\left(\beta_1,\beta_1\right)}\beta_1-\frac{\left(\alpha_s,\beta_2\right)}{\left(\beta_2,\beta_2\right)}\beta_2-\cdots -\frac{\left(\alpha_s,\beta_{s-1}\right)}{\left(\beta_{s-1},\beta_{s-1}\right)}\beta_{s-1};\\ \end{array} \end{equation*}
2. 再单位化，令

   \begin{equation*} \gamma_i=\frac{1}{\|\beta_i\|}\beta_i,\ i=1,2,\cdots ,s, \end{equation*}

   则得标准正交向量组\(\gamma_1,\gamma_2,\cdots ,\gamma_s\)满足

   \begin{equation*} \langle\alpha_1,\cdots ,\alpha_r\rangle=\langle\gamma_1,\cdots ,\gamma_r\rangle ,r=1,2,\cdots ,s \mbox{。} \end{equation*}

• 设\(\xi_1,\xi_2,\cdots ,\xi_n\)和\(\eta_1,\eta_2,\cdots,\eta_n\)是\(n\)维内积空间\(V\)的两个标准正交基，\(Q\)是从标准正交基\(\xi_1,\xi_2,\cdots ,\xi_n\)到标准正交基\(\eta_1,\eta_2,\cdots,\eta_n\)的过渡矩阵， 即

  \begin{equation*} (\eta_1,\eta_2,\cdots,\eta_n)=(\xi_1,\xi_2,\cdots ,\xi_n)Q, \end{equation*}
• 记\(Q=\left(q_{ij}\right)\)，则\(Q\)是\(n\)阶复方阵且

  \begin{equation*} \overline{Q}^TQ=E\mbox{。} \end{equation*}