将矩阵化为史密斯标准型

青青子衿

\begin{align*}
&\qquad\qquad\qquad\qquad\qquad\begin{pmatrix}
\lambda+1&-1&\\
4&\lambda-3&\\
-1&&\lambda-2\\
\end{pmatrix}\\
\\
&
\xrightarrow[]{\quad\,c_2+c_1\quad}
\begin{pmatrix}
\lambda&-1&\\
\lambda+1&\lambda-3&\\
-1&&\lambda-2\\
\end{pmatrix}
\xrightarrow[]{\quad-r_2+r_1\quad}
\begin{pmatrix}
-1&2-\lambda&\\
\lambda+1&\lambda-3&\\
-1&&\lambda-2\\
\end{pmatrix}\\
\\
&
\xrightarrow[-r_1+r_3]{\quad\left(\lambda+1\right)r_1+r_2\quad}
\begin{pmatrix}
-1&2-\lambda&\\
&\lambda\left(2-\lambda\right)-1&\\
&\lambda-2&\lambda-2\\
\end{pmatrix}
\xrightarrow[]{\quad\left(2-\lambda\right)c_1+c_2\quad}
\begin{pmatrix}
-1&&\\
&\lambda\left(2-\lambda\right)-1&\\
&\lambda-2&\lambda-2\\
\end{pmatrix}\\
\\
&
\xrightarrow[]{\quad\lambda\cdot\!\,r_3+r_2\quad}
\begin{pmatrix}
-1&&\\
&-1&\lambda\left(\lambda-2\right)\\
&\lambda-2&\lambda-2\\
\end{pmatrix}
\xrightarrow[]{\quad\left(\lambda-2\right)r_2+r_3\quad}
\begin{pmatrix}
-1&&\\
&-1&\lambda\left(\lambda-2\right)\\
&&\left(\lambda-1\right)^2\left(\lambda-2\right)\\
\end{pmatrix}\\
\\
&
\xrightarrow[]{\quad\lambda\left(\lambda-2\right)c_2+c_3\quad}
\begin{pmatrix}
-1&&\\
&-1&\\
&&\left(\lambda-1\right)^2\left(\lambda-2\right)\\
\end{pmatrix}
\xrightarrow[\quad\,c_2\cdot\left(-1\right)\quad]{\quad\,c_1\cdot\left(-1\right)\quad}
\begin{pmatrix}
1&&\\
&1&\\
&&\left(\lambda-1\right)^2\left(\lambda-2\right)\\
\end{pmatrix}\\
\end{align*}

\begin{align*}
\color{black}{
\begin{pmatrix}
-2&-2&4\\
-6&0&6\\
-7&-3&9\\
\end{pmatrix}

\qquad\qquad
\begin{pmatrix}
2&0&0\\
6&4&-2\\
3&1&1\\
\end{pmatrix}

\qquad\qquad
\begin{pmatrix}
5&-2&1\\
8&-1&2\\
7&0&3\\
\end{pmatrix}}
\end{align*}