高等代数教学辅导

\begin{equation*} Q^{-1}AQ={\rm diag}\left( \begin{pmatrix} \cos\theta_1 & -\sin\theta_1\\ \sin\theta_1 & \cos\theta_1\end{pmatrix}, \dots , \begin{pmatrix} \cos\theta_l & -\sin\theta_l\\ \sin\theta_l & \cos\theta_l\end{pmatrix}, E_r, -E_s\right), \end{equation*}

\begin{equation*} \det A = \prod\limits_{i=1}^l \begin{vmatrix} \cos\theta_i & -\sin\theta_i\\ \sin\theta_i & \cos\theta_i\end{vmatrix}\cdot 1^r\cdot (-1)^s=(-1)^s, \end{equation*}

\begin{equation} A = Q^{-1}\begin{pmatrix} E_r & {\bf 0}\\ {\bf 0} & -E_s\end{pmatrix}Q. \tag{8.3.1} \end{equation}

\begin{equation*} A^T=Q^T\begin{pmatrix} E_r & {\bf 0}\\ {\bf 0} & -E_s\end{pmatrix}Q=A, \end{equation*}

\begin{equation*} \begin{array}{cl} &\left(\phi(a\alpha+b \beta)-a\phi(\alpha)-b\phi(\beta),\phi(a\alpha+b \beta)-a\phi(\alpha)-b\phi(\beta)\right)\\ =&\left(\phi(a\alpha+b \beta),\phi(a\alpha+b \beta)\right)-2a\left(\phi(\alpha),\phi(a\alpha+b \beta)\right)\\ &-2b\left(\phi(\beta),\phi(a\alpha+b \beta)\right)+a^2\left(\phi(\alpha),\phi(\alpha)\right)\\ &+2ab\left(\phi(\alpha),\phi(\beta)\right)+b^2\left(\phi(\beta),\phi(\beta)\right)\\ =&\left(a\alpha+b \beta,a\alpha+b \beta\right)-2a\left(\alpha,a\alpha+b \beta\right)-2b\left(\beta,a\alpha+b \beta\right)\\ &+a^2\left(\alpha,\alpha\right)+2ab\left(\alpha,\beta\right)+b^2\left(\beta,\beta\right)\\ =&0 \end{array} \end{equation*}

\begin{equation*} \phi(a\alpha+b \beta)=a\phi(\alpha)+b\phi(\beta). \end{equation*}

\begin{equation*} (\phi(\alpha),\beta)=(\alpha,\phi^{-1}(\beta)), \end{equation*}

\begin{equation*} (\phi(\alpha),\beta)=(\alpha,\phi^{-1}(\beta))=0, \end{equation*}

\begin{equation*} U^\perp =\langle \xi_{r+1},\dots ,\xi_n\rangle. \end{equation*}

\begin{equation*} (\phi(\xi_i),\phi(\xi_j))=(\xi_i,\xi_j)=0 \end{equation*}

\begin{equation*} U^\perp =\langle \xi_{r+1},\dots ,\xi_n\rangle. \end{equation*}

\begin{equation*} \phi(\xi_1,\dots ,\xi_n)=(\xi_1,\dots ,\xi_n)\begin{pmatrix} A&B\\{\bf 0}&C \end{pmatrix}. \end{equation*}

\begin{equation*} \begin{pmatrix} A^{-1}&-A^{-1}BC^{-1}\\{\bf 0}&C^{-1} \end{pmatrix}=\begin{pmatrix} A^T&{\bf 0}\\B^T&C^T \end{pmatrix}, \end{equation*}

\begin{equation*} \phi(\alpha)=\lambda \alpha,\phi(\beta)=\mu \beta\mbox{，且}\left|\lambda\right|=\left|\mu\right|=1\mbox{。} \end{equation*}

\begin{equation*} \lambda\overline{\mu}(\alpha,\beta)=(\alpha,\beta). \end{equation*}

\begin{equation*} \phi (\beta)=\beta-2(\alpha,\beta)\alpha, \end{equation*}

\begin{equation*} \begin{array}{ccl}\phi(\xi)&=&\xi-2(\frac{\xi- \eta}{\|\xi-\eta\|},\xi)\frac{\xi- \eta}{\|\xi-\eta\|}\\ &=&\xi-2\frac{(\xi-\eta,\xi)}{(\xi-\eta,\xi-\eta)}(\xi-\eta)\\ &=&\xi-2\frac{1-(\xi,\eta)}{2-2(\xi,\eta)}(\xi-\eta)\\ &=&\eta . \end{array} \end{equation*}

\begin{equation*} \det A = (\det B)^2 = (\pm 1)^2=1. \end{equation*}

\begin{equation*} A=Q^{-1}{\rm diag}\left( \begin{pmatrix} \cos\theta_1 & -\sin\theta_1\\ \sin\theta_1 & \cos\theta_1\end{pmatrix}, \dots , \begin{pmatrix} \cos\theta_l & -\sin\theta_l\\ \sin\theta_l & \cos\theta_l\end{pmatrix}, E_r, -E_s\right)Q. \end{equation*}

\begin{equation*} \begin{pmatrix} -1 & 0\\ 0 & -1 \end{pmatrix}=\begin{pmatrix} 0 & -1\\ 1 & 0\end{pmatrix}^2, \end{equation*}

\begin{equation*} \phi_i(\alpha_1)=a\phi_i(\xi_1)=a \xi_1=\alpha_1, \end{equation*}

\begin{equation*} \begin{array}{ccl} \phi_i(\alpha)&=&\phi_i(\alpha_1)+\phi_i(\alpha_2)=\alpha_1+\alpha_2-2(\xi_i,\alpha_2)\xi_i\\ &=&\alpha-2(\xi_i,\alpha_1+\alpha_2)\xi_i=\alpha-2(\xi_i,\alpha)\xi_i,\end{array} \end{equation*}

\begin{equation*} \begin{array}{ccl} \phi_2\cdots\phi_s(\alpha)&=&\phi_2\cdots\phi_s(\alpha_1)+\phi_2\cdots\phi_s(\alpha_2)=\alpha_1+\phi_2\cdots\phi_s(\alpha_2)\\&=&\phi_1\phi(\alpha_1)+\phi_1\phi(\alpha_2)=\phi_1\phi(\alpha),\end{array} \end{equation*}

\begin{equation*} \phi=\phi_1\phi_2\cdots\phi_s \end{equation*}

\begin{equation*} \left(\alpha_i,\alpha_j\right)=\left(\phi(\beta_i),\phi(\beta_j)\right)=\left(\beta_i,\beta_j\right),\ i,j=1,2,\cdots ,m. \end{equation*}

\begin{equation*} \det G(\alpha_1,\dots ,\alpha_r)\neq 0, \end{equation*}

\begin{equation*} \det G(\beta_1,\dots ,\beta_r)=\det G(\alpha_1,\dots ,\alpha_r)\neq 0, \end{equation*}

\begin{equation*} \begin{array}{ll} & \left(\beta_i-\sum\limits_{j=1}^ra_{ji}\beta_j,\beta_i-\sum\limits_{j=1}^ra_{ji}\beta_j\right)\\ =&\left(\alpha_i-\sum\limits_{j=1}^ra_{ji}\alpha_j,\alpha_i-\sum\limits_{j=1}^ra_{ji}\alpha_j\right)\\ &=0,\end{array} \end{equation*}

\begin{equation*} \begin{array}{ccl} (\phi_1(\alpha),\phi_1(\beta))&=&\left(\sum\limits_{k=1}^r a_k \phi_1(\alpha_k),\sum\limits_{l=1}^r b_l \phi_1(\alpha_l)\right)\\ &=&\sum\limits_{k=1}^r\sum\limits_{l=1}^r a_kb_l\left(\phi_1(\alpha_k),\phi_1(\alpha_l)\right)\\ &=&\sum\limits_{k=1}^r\sum\limits_{l=1}^r a_kb_l\left(\beta_k,\beta_l\right)\\ &=&\sum\limits_{k=1}^r\sum\limits_{l=1}^r a_kb_l\left(\alpha_k,\alpha_l\right)\\ &=&\left(\sum\limits_{k=1}^r a_k \alpha_k,\sum\limits_{l=1}^r b_l \alpha_l\right)\\ &=&(\alpha,\beta) \end{array} \end{equation*}

\begin{equation*} \phi_2:U_1^\bot\rightarrow U_2^\bot \end{equation*}

\begin{equation*} \phi(\alpha+\beta)=\phi_1(\alpha)+\phi_2(\beta),\ \forall \alpha\in U_1,\ \beta\in U_1^\bot . \end{equation*}

\begin{equation*} \begin{array}{ccl} \left(\phi(X),\phi(Y)\right)&=&\left(\phi_1(X_1)+\phi_2(X_2),\phi_1(Y_1)+\phi_2(Y_2)\right)\\ &=&\left(\phi_1(X_1),\phi_1(Y_1)\right)+\left(\phi_1(X_1),\phi_2(Y_2)\right)\\ &&+\left(\phi_2(X_2),\phi_1(Y_1)\right)+\left(\phi_2(X_2),\phi_2(Y_2)\right)\\ &=&\left(\phi_1(X_1),\phi_1(Y_1)\right)+\left(\phi_2(X_2),\phi_2(Y_2)\right)\\ &=&(X_1,Y_1)+(X_2,Y_2)\\ &=&(X_1,Y_1)+(X_1,Y_2)+(X_2,Y_1)+(X_2,Y_2)\\ &=&(X,Y), \end{array} \end{equation*}

\begin{equation*} \begin{array}{ll} \phi(\alpha_i)&=\phi_1(\alpha_i)\\ &=\phi_1(a_{1i}\alpha_1+\cdots +a_{ri}\alpha_r)\\ &=a_{1i}\beta_1+\dots +a_{ri}\beta_r\\ &=\beta_i.\end{array} \end{equation*}