高等代数教学辅导

\begin{equation*} \begin{array}{cl} &\begin{pmatrix} F\left((\lambda-2)^2(\lambda^2+2)\right)&0\\ 0&F\left((\lambda-2)^2(\lambda^2+2)^2\right) \end{pmatrix}\\ =&\begin{pmatrix} F\left(\lambda^4-4\lambda^3+6\lambda^2-8\lambda+8\right)&0\\ 0&F\left(\lambda^6-4\lambda^5+8\lambda^4-16\lambda^3+20\lambda^2-16\lambda+16\right) \end{pmatrix}, \end{array} \end{equation*}

\begin{equation*} \begin{pmatrix} 0&0&0&-8&0&0&0&0&0&0\\ 1&0&0&8&0&0&0&0&0&0\\ 0&1&0&-6&0&0&0&0&0&0\\ 0&0&1&4&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&-16\\ 0&0&0&0&1&0&0&0&0&16\\ 0&0&0&0&0&1&0&0&0&-20\\ 0&0&0&0&0&0&1&0&0&16\\ 0&0&0&0&0&0&0&1&0&-8\\ 0&0&0&0&0&0&0&0&1&4 \end{pmatrix}. \end{equation*}

\begin{equation*} g_1(\lambda)=1,g_2(\lambda)=\lambda-2,g_3(\lambda)=(\lambda-2)^2, \end{equation*}

\begin{equation*} \begin{pmatrix} F(\lambda-2)&0\\ 0&F\left((\lambda-2)^2\right) \end{pmatrix}=\begin{pmatrix} 2&0&0\\ 0&0&-4\\ 0&1&4 \end{pmatrix}. \end{equation*}

\begin{equation*} \lambda E-B\simeq \begin{pmatrix} 1&0\\ 0&(\lambda-1)^2 \end{pmatrix},\quad \lambda E-C\simeq \begin{pmatrix} 1&0&0\\ 0&\lambda-2&0\\ 0&0&(\lambda-2)^2 \end{pmatrix}, \end{equation*}

\begin{equation*} \lambda E-A\simeq \begin{pmatrix} 1&0&0&0&0\\ 0&(\lambda-1)^2&0&0&\\ 0&0&1&0&0\\ 0&0&0&\lambda-2&0\\ 0&0&0&0&(\lambda-2)^2 \end{pmatrix}. \end{equation*}

\begin{equation*} D_1(\lambda)=D_2(\lambda)=D_3(\lambda)=1,D_4(\lambda)=\lambda-2,D_5(\lambda)=(\lambda-1)^2(\lambda-2)^3. \end{equation*}

\begin{equation*} g_1(\lambda)=g_2(\lambda)=g_3(\lambda)=1,g_4(\lambda)=\lambda-2,g_5(\lambda)=(\lambda-1)^2(\lambda-2)^2. \end{equation*}

\begin{equation*} \begin{pmatrix} F(\lambda-2)&0\\ 0&F\left((\lambda-1)^2(\lambda-2)^2\right) \end{pmatrix}, \end{equation*}

\begin{equation*} \begin{pmatrix} 2&0&0&0&0\\ 0&0&0&0&-4\\ 0&1&0&0&12\\ 0&0&1&0&-13\\ 0&0&0&1&6 \end{pmatrix}. \end{equation*}

\begin{equation*} \lambda-1, (\lambda-1)^3, \lambda^2+1, (\lambda^2+1)^2 , \lambda^2-2, \end{equation*}

\begin{equation*} \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{pmatrix}. \end{equation*}

\begin{equation*} \lambda-1, (\lambda-1)^3, \lambda+i,\lambda-i, (\lambda+i)^2 , (\lambda-i)^2 , \lambda+\sqrt{2}, \lambda-\sqrt{2}, \end{equation*}

\begin{equation*} \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -i & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & i & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -i & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -i & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & i & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & i & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{2} \end{pmatrix}. \end{equation*}

\begin{equation*} g_1(\lambda)=\cdots =g_{n-1}(\lambda)=1,g_n(\lambda)=m_A(\lambda). \end{equation*}

\begin{equation*} 1,1,1,(\lambda+1)^2(\lambda^2+\lambda+1)\ \ \mbox{和}\ \ 1,1,1,1,(\lambda+1)(\lambda^2+\lambda+1)^2, \end{equation*}

\begin{equation*} \begin{array}{ll} \chi_{F(f(\lambda))} & =\begin{vmatrix} \lambda&0&\cdots&0&a_0\\ -1&\lambda&\cdots&0&a_1\\ 0&-1&\ddots&\vdots&\vdots\\ \vdots&\ddots&\ddots&\lambda& a_{r-2} \\ 0&\cdots& 0 &-1&\lambda+a_{r-1}\\ \end{vmatrix}\\ & =\lambda^r+a_{r-1}\lambda^{r-1}+\cdots +a_1\lambda+a_0\\ & =f(\lambda),\end{array} \end{equation*}

\begin{equation*} m_{F(f(\lambda))}(\lambda)=\lambda^s + b_{s-1}\lambda^{s-1} + \dots + b_1\lambda + b_0, \end{equation*}

\begin{equation*} F(f(\lambda))^s\varepsilon_1 + b_{s-1}F(f(\lambda))^{s-1}\varepsilon_1 + \dots + b_1F(f(\lambda))\varepsilon_1 + b_0\varepsilon_1 = {\bf 0}. \end{equation*}

\begin{equation*} \varepsilon_{s+1} + b_{s-1}\varepsilon_{s} + \dots + b_1\varepsilon_ + b_0\varepsilon_1={\bf 0}, \end{equation*}

\begin{equation*} 1,\dots,1,m_A⁡(\lambda)=\lambda^n+a_{n-1}\lambda^{n-1}+\dots+a_0, \end{equation*}

\begin{equation*} P^{-1}AP=F(m_A(\lambda))=\begin{pmatrix} 0 & & & -a_0\\ 1 & 0 & & -a_1\\ & \ddots & \ddots & \vdots\\ & & 1 & -a_{n-1} \end{pmatrix}. \end{equation*}

\begin{equation*} F(m_A(\lambda))^jC=CF(m_A(\lambda))^j. \end{equation*}

\begin{equation*} C\varepsilon_1=(c_0E_n+c_1F(m_A(\lambda))+\dots+c_{n-1}F(m_A(\lambda))^{n-1})\varepsilon_1. \end{equation*}

\begin{equation*} \begin{array}{ll} C\varepsilon_k & = C(F(m_A(\lambda))^{k-1}\varepsilon_1)\\ & = F(m_A(\lambda))^{k-1}(C\varepsilon_1)\\ & = (c_0E_n+c_1F(m_A(\lambda))+\dots+c_{n-1}F(m_A(\lambda))^{n-1})F(m_A(\lambda))^{k-1}\varepsilon_1\\ & = (c_0E_n+c_1F(m_A(\lambda))+\dots+c_{n-1}F(m_A(\lambda))^{n-1})\varepsilon_k, \end{array} \end{equation*}

\begin{equation*} C=c_0E_n+c_1F(m_A(\lambda))+\dots+c_{n-1}F(m_A(\lambda))^{n-1}, \end{equation*}

\begin{equation*} B=PCP^{-1}=\sum\limits_{i=0}^{n-1}c_{i}PF(m_A(\lambda))^{i}P^{-1}=\sum\limits_{i=0}^{n-1}c_{i}A^i. \end{equation*}