构造分块矩阵 \(\begin{pmatrix}
\lambda E_3-A & E_3 \\
E_3 & \end{pmatrix}\),对此分块矩阵的前3行和前3列进行\(\lambda\)-矩阵初等变换,把\(\lambda E_3-A\)变成其Smith标准形\(\Lambda_A\),记获得的分块矩阵为
\begin{equation*}
\begin{pmatrix}
\Lambda_A & M_1(\lambda) \\
N_1(\lambda) & \end{pmatrix},
\end{equation*}
计算可知
\begin{equation*}
M_1(\lambda) = \begin{pmatrix}
0 & \frac{1}{9} & 0 \\
0 & 1 & -1 \\
-9 & 3 & \lambda - 7
\end{pmatrix},\quad N_1(\lambda)= \begin{pmatrix}
1 & -\frac{1}{9} \lambda + \frac{4}{9} & -\frac{1}{9} \lambda - \frac{2}{9} \\
0 & 1 & 1 \\
0 & 0 & 1
\end{pmatrix}.
\end{equation*}
可以验证
\begin{equation*}
M_1(\lambda)(\lambda E_3 - A)N_1(\lambda) = \Lambda_A = \begin{pmatrix}
1 & 0 & 0 \\
0 & \lambda-1 & 0 \\
0 & 0 & (\lambda-1)^2
\end{pmatrix}.
\end{equation*}
类似方法将\(\lambda E_3-J\)变成其Smith标准形,求得
\begin{equation*}
M_2(\lambda) = \begin{pmatrix}
0 & 0 & -1 \\
-1 & 0 & -\lambda + 1 \\
0 & 1 & \lambda - 1
\end{pmatrix},\quad N_2(\lambda)= \begin{pmatrix}
1 & -1 & 0 \\
1 & 0 & \lambda - 1 \\
0 & 0 & 1
\end{pmatrix},
\end{equation*}
使得
\begin{equation*}
M_2(\lambda)(\lambda E_3 - J)N_2(\lambda) = \Lambda_J=\Lambda_A.
\end{equation*}
取
\begin{equation*}
M(\lambda) = M_2^{-1}(\lambda)M_1(\lambda) = \begin{pmatrix}
0 & \frac{1}{9}\lambda - \frac{10}{9} & 1 \\
-9 & \frac{1}{9} \lambda + \frac{26}{9} & \lambda - 7 \\
0 & -\frac{1}{9} & 0
\end{pmatrix},
\end{equation*}
\begin{equation*}
N(\lambda) = N_1(\lambda)N_2^{-1}(\lambda)=\begin{pmatrix}
\frac{1}{9} \lambda - \frac{4}{9} & -\frac{1}{9} \lambda + \frac{13}{9} & \frac{1}{9} \lambda^{2} - \frac{5}{3} \lambda + \frac{11}{9} \\
-1 & 1 & -\lambda + 2 \\
0 & 0 & 1
\end{pmatrix},
\end{equation*}
可以验证
\begin{equation*}
M(\lambda)(\lambda E_3 - A)N(\lambda) = \lambda E_3 - J.
\end{equation*}
将\(M(\lambda)\)写成系数为矩阵的多项式
\begin{equation*}
M(\lambda) = \begin{pmatrix}
0 & \frac{1}{9} & 1 \\
-9 & \frac{1}{9} & -7 \\
0 & -\frac{1}{9} & 0
\end{pmatrix} + \lambda \begin{pmatrix}
0 & -\frac{10}{9} & 0 \\
0 & \frac{26}{9} & 1 \\
0 & 0 & 0
\end{pmatrix},
\end{equation*}
用\(J\)替代上式中的\(\lambda\),得到\(M(\lambda)\) 左除\(\lambda E_3 - J\)的余式\(L=\begin{pmatrix}
0 & -1 & 1 \\
-9 & 3 & -6 \\
0 & 0 & 1
\end{pmatrix}\),则\(L^{-1}\)就是我们要求的过渡矩阵\(P\)。可以验证\(P^{-1}AP=J\)。