高等代数教学辅导

\begin{equation*} \begin{array}{ll}\begin{pmatrix} A\\E \end{pmatrix}&=\begin{pmatrix} 1&-1&1\\ -1&1&3\\ 1&3&1\\ 1&0&0\\ 0&1&0\\ 0&0&1 \end{pmatrix}\xrightarrow[c_2+c_1]{r_2+r_1}\begin{pmatrix} 1&0&1\\ 0&0&4\\ 1&4&1\\ 1&1&0\\ 0&1&0\\ 0&0&1 \end{pmatrix}\\ &\xrightarrow[c_3-c_1]{r_3-r_1}\begin{pmatrix} 1&0&0\\ 0&0&4\\ 0&4&0\\ 1&1&-1\\ 0&1&0\\ 0&0&1 \end{pmatrix}\xrightarrow[c_2+c_3]{r_2+r_3}\begin{pmatrix} 1&0&0\\ 0&8&4\\ 0&4&0\\ 1&0&-1\\ 0&1&0\\ 0&1&1 \end{pmatrix}\\ &\xrightarrow[c_3-\frac{1}{2}c_2]{r_3-\frac{1}{2}r_2}\begin{pmatrix} 1&0&0\\ 0&8&0\\ 0&0&-2\\ 1&0&-1\\ 0&1&-\frac{1}{2}\\ 0&1&\frac{1}{2} \end{pmatrix},\end{array} \end{equation*}

\begin{equation*} C=E(1,2(1))E(1,3(-1))E(3,2(1))E(2,3(-\frac{1}{2}))=\begin{pmatrix} 1&0&-1\\ 0&1&-\frac{1}{2}\\ 0&1&\frac{1}{2} \end{pmatrix}, \end{equation*}

\begin{equation*} f(x_1,x_2,x_3)=g(y_1,y_2,y_3)=y_1^2+8y_2^2-2y_3^2. \end{equation*}

\begin{equation*} y_1^2+8y_2^2-2y_3^2, \end{equation*}

\begin{equation*} Q=C \begin{pmatrix} 1&0&0\\ 0&2\sqrt{2}&0\\ 0&0&\sqrt{2} \end{pmatrix}^{-1}=\begin{pmatrix} 1&0&-\frac{\sqrt{2}}{2}\\ 0&\frac{\sqrt{2}}{4}&-\frac{\sqrt{2}}{4}\\ 0&\frac{\sqrt{2}}{4}&\frac{\sqrt{2}}{4} \end{pmatrix}, \end{equation*}

\begin{equation*} z_1^2+z_2^2-z_3^2. \end{equation*}

\begin{equation*} C=\begin{pmatrix} 1 & 1 & & & & &\\ 1 & -1 & & & & &\\ & & 1 & 1 & & &\\ & & 1 & -1 & & &\\ & & & & \ddots & &\\ & & & & & 1 & 1\\ & & & & & 1 & -1 \end{pmatrix}, \end{equation*}

\begin{equation*} y_1^2-y_2^2+y_3^2-y_4^2+\dots+y_{2n-1}^2+y_{2n}^2. \end{equation*}

\begin{equation*} Y^(C^TAC)Y = y_1^2+\cdots+y_p^2-y_{p+1}^2-\cdots -y_r^2. \end{equation*}

\begin{equation*} \alpha_3=C(\varepsilon_1+\varepsilon_{p+1})\in\mathbb{R}^n, \end{equation*}

\begin{equation*} \begin{array}{ll} \alpha_3^TA \alpha_3 & =(\varepsilon_1+\varepsilon_{p+1})^T(C^TAC)(\varepsilon_1+\varepsilon_{p+1})\\ & =1^2+0^2+\cdots+0^2-1^2-0^2+\cdots-0^2\\ & =0. \end{array} \end{equation*}

\begin{equation*} \left\{\begin{array}{l} x_1=u_1-u_2-3u_3,\\ x_2=u_2+u_3,\\ x_3=u_2-u_3, \end{array}\right. \end{equation*}

\begin{equation*} f_2(y_1,y_2,y_3)=(y_1+2y_2-y_3)^2-2y_2^2-2y_3^2, \end{equation*}

\begin{equation*} f_3(z_1,z_2,z_3)=-4(z_1+\frac{1}{2}z_2-\frac{1}{2}z_3)^2+16z_2z_3, \end{equation*}

\begin{equation*} \left\{\begin{array}{l} z_1=v_1-v_3,\\ z_2=v_2+v_3,\\ z_3=v_2-v_3, \end{array}\right. \end{equation*}

\begin{equation*} -4v_1^2+16v_2^2-16v_3^2, \end{equation*}

\begin{equation*} \sum\limits_{r=0}^n (r+1)=\frac{(n+1)(n+2)}{2}. \end{equation*}

\begin{equation*} A=\begin{pmatrix} 0&\frac{a}{2}&\frac{b}{2}\\ \frac{a}{2}&0&\frac{c}{2}\\ \frac{b}{2}&\frac{c}{2}&0 \end{pmatrix}. \end{equation*}

\begin{equation*} \lambda_1+\lambda_2+\lambda_3={\rm tr}(A)=0,\ \lambda_1 \lambda_2 \lambda_3=\det A=\frac{abc}{4}. \end{equation*}

\begin{equation*} X^TAX=(a_1x_1+\dots +a_nx_n)(b_1x_1+\dots +b_nx_n), \end{equation*}

\begin{equation*} y_1^2+\cdots+y_p^2-y_{p+1}^2-\cdots -y_{p+q}^2. \end{equation*}

\begin{equation*} p+q=n,\ p-q=s, \end{equation*}

\begin{equation*} \frac{1}{2}(n-|s|)=\frac{1}{2}(p+q-|p-q|)={\rm min}(p,q). \end{equation*}

\begin{equation*} V=\langle C(\varepsilon_1+\varepsilon_{p+1}), C(\varepsilon_2+\varepsilon_{p+1}), \dots , C(\varepsilon_p+\varepsilon_{p+1}) \rangle, \end{equation*}

\begin{equation*} \begin{array}{ll} & f(C(\varepsilon_i+\varepsilon_{p+1}))\\ = & 0^2+\dots+0^2+1^2+0^2+\dots+0^2-1^2-0^2-\dots-0^2\\ = & 0, \end{array} \end{equation*}

\begin{equation*} f(x_1,\ldots,x_n)=y_1^2+\cdots +y_p^2-y_{p+1}^2-\cdots -y_{p+q}^2, \end{equation*}

\begin{equation} \begin{array}{ll} & y_1^2+\cdots +y_p^2-y_{p+1}^2-\cdots -y_{p+q}^2\\ = & z_1^2+\cdots+z_k^2-z_{k+1}^2-\cdots-z_{k+s}^2. \end{array}\tag{9.2.1} \end{equation}

\begin{equation*} \left\{\begin{array}{l} a_{11}x_1+a_{12}x_2+\cdots +a_{1n}x_n=0,\\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ a_{k1}x_1+a_{k2}x_2+\cdots +a_{kn}x_n=0,\\ b_{p+1,1}x_1+b_{p+1,2}x_2+\cdots +b_{p+1,n}x_n=0,\\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ b_{n1}x_1+b_{n2}x_2+\cdots +b_{nn}x_n=0, \end{array}\right. \end{equation*}

\begin{equation*} C^{-1}\alpha=(b_1,\dots ,b_p,0,\dots ,0)\neq {\bf 0}, \end{equation*}

\begin{equation*} A \alpha=(0,\dots ,0,a_{k+1},\dots ,a_n)\neq {\bf 0}. \end{equation*}

\begin{equation} X^TAX=y_1^2+\dots+y_a^2-y_{a+1}^2-\dots-y_r^2,\tag{9.2.2} \end{equation}

\begin{equation} X^TBX=z_1^2+\dots+z_b^2-z_{b+1}^2-\dots-z_s^2,\tag{9.2.3} \end{equation}

\begin{equation} X^T(A+B)X=w_1^2+\dots+w_c^2-w_{c+1}^2-\dots-w_t^2.\tag{9.2.4} \end{equation}

\begin{equation*} \left\{\begin{array}{l} p_{11}x_1+p_{12}x_2+\cdots +p_{1n}x_n=0,\\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ p_{a1}x_1+p_{a2}x_2+\cdots +p_{an}x_n=0,\\ q_{11}x_1+q_{12}x_2+\cdots +q_{1n}x_n=0,\\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ q_{b1}x_1+q_{b2}x_2+\cdots +q_{bn}x_n=0,\\ r_{c+1,1}x_1+r_{c+1,2}x_2+\cdots +r_{c+1,n}x_n=0,\\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ r_{n1}x_1+r_{n2}x_2+\cdots +r_{nn}x_n=0, \end{array}\right. \end{equation*}

\begin{equation*} P^{-1}\alpha=(0,\dots ,0,y_{a+1},\dots ,y_n), \end{equation*}

\begin{equation*} Q^{-1} \alpha=(0,\dots ,0,z_{b+1},\dots ,z_n). \end{equation*}

\begin{equation} \alpha^T(A+B)\alpha=-y_{a+1}^2-\dots-y_r^2-z_{b+1}^2-\dots-z_s^2\leq 0.\tag{9.2.5} \end{equation}

\begin{equation*} \alpha^T(A+B)\alpha=w_1^2+\dots+w_c^2 >0, \end{equation*}

\begin{equation*} \begin{pmatrix} E_n&0\\A^{-1}&E _n \end{pmatrix}^T B \begin{pmatrix} E_n&0\\A^{-1}&E_n \end{pmatrix}=\begin{pmatrix} 2E_n&A\\A^T&0 \end{pmatrix}, \end{equation*}

\begin{equation*} \begin{pmatrix} E_n&-\frac{1}{2}A\\0&E_n \end{pmatrix}^T\begin{pmatrix} 2E_n&A\\A^T&0 \end{pmatrix}\begin{pmatrix} E_n&-\frac{1}{2}A\\0&E_n \end{pmatrix}=\begin{pmatrix} 2E_n&0\\0&-\frac{1}{2}A^TA \end{pmatrix}, \end{equation*}

\begin{equation*} \begin{pmatrix} \lambda E_n&-A\\-A^T &\lambda E_n \end{pmatrix} \begin{pmatrix} E_n&A\\0&\lambda E_n \end{pmatrix}= \begin{pmatrix} \lambda E_n&0\\-A^T&\lambda^2E_n-A^TA \end{pmatrix}, \end{equation*}

\begin{equation*} f_B(\lambda)=\begin{vmatrix} \lambda E_n&-A\\-A^T &\lambda E_n \end{vmatrix}=\det (\lambda^2E_n-A^TA). \end{equation*}

\begin{equation*} \lambda_1,\lambda_2,\cdots ,\lambda_n\in\mathbb{R}^+, \end{equation*}

\begin{equation*} \pm\sqrt{\lambda_1},\pm\sqrt{\lambda_2},\cdots ,\pm\sqrt{\lambda_n}, \end{equation*}

\begin{equation*} f (X) =X^T\frac{A + A^{T}}{2}X. \end{equation*}

\begin{equation*} b_{11}= b_{22}= \dots = b_{nn}= a, \end{equation*}

\begin{equation*} \sum\limits_{j=1}^{n}|b_{ij}| = \frac{1}{2}\sum\limits_{j=1}^{n}\left| a_{ij}+ a_{ji}\right| \leq \frac{1}{2}\left(\sum\limits_{j=1}^{n}\left| a_{ij}\right| + \sum\limits_{j=1}^{n}\left| a_{ji}\right| \right)< 2a, \end{equation*}

\begin{equation*} \sum\limits_{j \neq i} |b_{ij}| < 2a - |b_{ii}| =a. \end{equation*}

\begin{equation*} \lambda=\frac{\sum\limits_{j=1}^{n}b_{ij}a_j}{a_i}=a+\frac{\sum\limits_{j\neq i}b_{ij}a_j}{a_i}\geq a-\sum\limits_{j \neq i} |b_{ij}| >0, \end{equation*}