高等代数教学辅导

\begin{equation*} B(\lambda)=B_t\lambda^t+B_{t-1}\lambda^{t-1}+\cdots +B_1\lambda+B_0, \end{equation*}

\begin{equation*} \left(A_sB_t\right)\lambda^{s+t}+\left(A_sB_{t-1}+A_{s-1}B_t\right)\lambda^{s+t-1}+\cdots +\left(A_1B_0+A_0B_1\right)\lambda+A_0B_0=E. \end{equation*}

\begin{equation*} A_sB_t=0,A_0B_0=E. \end{equation*}

\begin{equation*} B(\lambda)=P(\lambda)A(\lambda)Q(\lambda),\ D(\lambda)=R(\lambda)C(\lambda)S(\lambda)\mbox{。} \end{equation*}

\begin{equation*} U(\lambda)=\begin{pmatrix} P(\lambda)&0\\0&R(\lambda) \end{pmatrix},V(\lambda)=\begin{pmatrix} Q(\lambda)&0\\0&S(\lambda) \end{pmatrix}, \end{equation*}

\begin{equation*} \begin{pmatrix} B(\lambda)&0\\0&D(\lambda) \end{pmatrix}=U(\lambda)\begin{pmatrix} A(\lambda)&0\\0&C(\lambda) \end{pmatrix}V(\lambda)\mbox{。} \end{equation*}

\begin{equation*} \begin{pmatrix} A(\lambda)&0\\0&C(\lambda) \end{pmatrix}\simeq \begin{pmatrix} B(\lambda)&0\\0&D(\lambda) \end{pmatrix}\mbox{。} \end{equation*}

\begin{equation*} \begin{array}{l}A(\lambda)\stackrel{E(1,2(1))}{\longrightarrow}\begin{pmatrix} 1&\lambda^2+\lambda&0\\ \lambda&\lambda&-\lambda\\ 1+\lambda^2&\lambda^2&-\lambda^2 \end{pmatrix} \xrightarrow[E(1,2(-\lambda^2- \lambda))]{ }\begin{pmatrix} 1&0&0\\ \lambda&-\lambda^3- \lambda^2+\lambda&-\lambda\\ 1+\lambda^2&-\lambda^4- \lambda^3- \lambda&-\lambda^2 \end{pmatrix}\\ \xrightarrow{{\begin{array}{c} E(2,1(-\lambda))\\E(3,1(- \lambda^2-1)) \end{array}}}\begin{pmatrix} 1&0&0\\ 0&-\lambda^3- \lambda^2+\lambda&-\lambda\\ 0&-\lambda^4- \lambda^3- \lambda&-\lambda^2 \end{pmatrix}\xrightarrow[E(2,3)]{}\begin{pmatrix} 1&0&0\\ 0&-\lambda&-\lambda^3- \lambda^2+\lambda\\ 0&-\lambda^2&-\lambda^4- \lambda^3- \lambda \end{pmatrix}\\ \xrightarrow{E(3,2(- \lambda))}\begin{pmatrix} 1&0&0\\ 0&-\lambda&-\lambda^3- \lambda^2+\lambda\\ 0&0&-\lambda^2- \lambda \end{pmatrix}\xrightarrow[E(2,3(- \lambda^2- \lambda+1))]{}\begin{pmatrix} 1&0&0\\ 0&-\lambda&0\\ 0&0&-\lambda^2- \lambda \end{pmatrix}\\ \xrightarrow{E(2(-1)),E(3(-1))}\begin{pmatrix} 1&0&0\\ 0&\lambda&0\\ 0&0&\lambda^2+\lambda \end{pmatrix} \end{array} \end{equation*}

\begin{equation*} \begin{array}{ccl}A(\lambda)&\xrightarrow[{\begin{array}{c} E(1,4)\\E(2,3) \end{array}}]{}&\begin{pmatrix} \lambda^2&0&0&0\\ 0&\lambda^2-\lambda&0&0\\ 0&0&(\lambda-1)^2&0\\ 0&0&0&\lambda^2-\lambda \end{pmatrix}\\ &\xrightarrow{E(1,2(1)) }&\begin{pmatrix} \lambda^2&\lambda^2-\lambda&0&0\\ 0&\lambda^2-\lambda&0&0\\ 0&0&(\lambda-1)^2&0\\ 0&0&0&\lambda^2-\lambda \end{pmatrix}\\ &\xrightarrow[E(1,2(-1))]{}&\begin{pmatrix} \lambda^2&-\lambda&0&0\\ 0&\lambda^2-\lambda&0&0\\ 0&0&(\lambda-1)^2&0\\ 0&0&0&\lambda^2-\lambda \end{pmatrix} \end{array} \end{equation*}

\begin{equation*} \begin{array}{cl} \xrightarrow[E(1,2)]{}&\begin{pmatrix} -\lambda&\lambda^2&0&0\\ \lambda^2-\lambda&0&0&0\\ 0&0&(\lambda-1)^2&0\\ 0&0&0&\lambda^2-\lambda \end{pmatrix}\\ \xrightarrow{E(1,3(\lambda-1))}&\begin{pmatrix} -\lambda&\lambda^2&(\lambda-1)^2&0\\ \lambda^2-\lambda&0&0&0\\ 0&0&(\lambda-1)^2&0\\ 0&0&0&\lambda^2-\lambda \end{pmatrix}\\ \xrightarrow[E(3,1(\lambda-2))]{}&\begin{pmatrix} -\lambda&\lambda^2&1&0\\ \lambda^2-\lambda&0&(\lambda^2- \lambda)(\lambda-2)&0\\ 0&0&(\lambda-1)^2&0\\ 0&0&0&\lambda^2-\lambda \end{pmatrix}\\ \xrightarrow[E(1,3)]{}&\begin{pmatrix} 1&\lambda^2&-\lambda&0\\ (\lambda^2- \lambda)(\lambda-2)&0&\lambda^2-\lambda&0\\ (\lambda-1)^2& 0&0&0\\ 0&0&0&\lambda^2-\lambda \end{pmatrix} \end{array} \end{equation*}

\begin{equation*} \begin{array}{cl} \xrightarrow[{\begin{array}{c} E(1,2(-\lambda^2))\\E(1,3(\lambda)) \end{array}}]{{\begin{array}{c} E(2,1(-(\lambda^2- \lambda)(\lambda-2)))\\E(3,1(-(\lambda-1)^2)) \end{array}}}&\begin{pmatrix} 1&0&0&0\\ 0&-\lambda^3(\lambda- 1)(\lambda-2)&\lambda (\lambda-1)^3&0\\ 0& -\lambda^2(\lambda-1)^2&\lambda(\lambda-1)^2&0\\ 0&0&0&\lambda^2-\lambda \end{pmatrix}\\ \xrightarrow[E(2,4)]{E(2,4)}&\begin{pmatrix} 1&0&0&0\\ 0&\lambda(\lambda- 1)&0&0\\ 0& 0&\lambda(\lambda-1)^2&-\lambda^2(\lambda-1)^2\\ 0&0&\lambda(\lambda-1)^3&-\lambda^3(\lambda-1)(\lambda-2) \end{pmatrix}\\ \xrightarrow{E(4,3(1- \lambda))}&\begin{pmatrix} 1&0&0&0\\ 0&\lambda(\lambda- 1)&0&0\\ 0& 0&\lambda(\lambda-1)^2&-\lambda^2(\lambda-1)^2\\ 0&0&0&\lambda^2(\lambda-1) \end{pmatrix}\\ \xrightarrow{E(3,4(1))}&\begin{pmatrix} 1&0&0&0\\ 0&\lambda(\lambda- 1)&0&0\\ 0& 0&\lambda(\lambda-1)^2&-\lambda^2(\lambda-1)(\lambda-2)\\ 0&0&0&\lambda^2(\lambda-1) \end{pmatrix} \end{array} \end{equation*}

\begin{equation*} \begin{array}{cl} \xrightarrow[E(3,4(\lambda-1))]{}&\begin{pmatrix} 1&0&0&0\\ 0&\lambda(\lambda- 1)&0&0\\ 0& 0&\lambda(\lambda-1)^2&\lambda(\lambda-1)\\ 0&0&0&\lambda^2(\lambda-1) \end{pmatrix}\\ \xrightarrow[E(3,4)]{}&\begin{pmatrix} 1&0&0&0\\ 0&\lambda(\lambda- 1)&0&0\\ 0& 0&\lambda(\lambda-1)&\lambda(\lambda-1)^2\\ 0&0&\lambda^2(\lambda-1) &0 \end{pmatrix}\\ \xrightarrow[E(3,4(1- \lambda))]{E(4,3(-\lambda))}&\begin{pmatrix} 1&0&0&0\\ 0&\lambda(\lambda- 1)&0&0\\ 0& 0&\lambda(\lambda-1)&0\\ 0&0&0&-\lambda^2(\lambda-1)^2 \end{pmatrix}\\ \xrightarrow{E(4(-1))}&\begin{pmatrix} 1&0&0&0\\ 0&\lambda(\lambda- 1)&0&0\\ 0& 0&\lambda(\lambda-1)&0\\ 0&0&0&\lambda^2(\lambda-1)^2 \end{pmatrix}. \end{array} \end{equation*}

\begin{equation*} \begin{pmatrix} f(\lambda)&0\\0&g(\lambda) \end{pmatrix}=P\ \begin{pmatrix} g(\lambda)&0\\0&f(\lambda) \end{pmatrix}P, \end{equation*}

\begin{equation*} \begin{array}{cl}\begin{pmatrix} f(\lambda)&0\\0&g(\lambda) \end{pmatrix}&\xrightarrow[E(2,1(v(\lambda) ))]{E(2,1(u(\lambda)))}\begin{pmatrix} f(\lambda)&0\\1&g(\lambda) \end{pmatrix}\stackrel{E(1,2)}{\longrightarrow}\begin{pmatrix} 1&g(\lambda)\\f(\lambda)&0 \end{pmatrix}\\&\xrightarrow[{\begin{array}{c} E(1,2(-g(\lambda)))\\E(2(-1)) \end{array}}]{{\begin{array}{c} E(2,1(-f(\lambda))) \end{array}}}\begin{pmatrix} 1&0\\0&f(\lambda)g(\lambda) \end{pmatrix}\mbox{。}\end{array} \end{equation*}