Answer
See below
Work Step by Step
1. Find eigenvalues:
(A-$\lambda$I)$\vec{V}$=$\vec{0}$
$\begin{bmatrix} -\lambda & 0 & 0 & 1 & 4\\ 0 & -\lambda & 1 & 1 & 1\\ 0 & 0 & -\lambda & 6 & 0\\ 0 & 0 & 0 & -\lambda & 0\\ 0 & 0 & 0 & 0 & -\lambda\end{bmatrix}\begin{bmatrix} v_1\\ v_2 \\ v_3 \\ v_4 \\ v_5 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \\0 \\0 \\ 0 \end{bmatrix}$
$\begin{bmatrix} -\lambda & 0 & 0 & 1 & 4\\ 0 & -\lambda & 1 & 1 & 1\\ 0 & 0 & -\lambda & 6 & 0\\ 0 & 0 & 0 & -\lambda & 0\\ 0 & 0 & 0 & 0 & -\lambda\end{bmatrix}=0$
$\lambda^5=0$
$\lambda_1=0$ with multiplicity 5.
2. Find eigenvectors:
For $\lambda=0$
let $B=A-\lambda_1I$
$B=\begin{bmatrix} -\lambda & 0 & 0 & 1 & 4\\ 0 & -\lambda & 1 & 1 & 1\\ 0 & 0 & -\lambda & 6 & 0\\ 0 & 0 & 0 & -\lambda & 0\\ 0 & 0 & 0 & 0 & -\lambda\end{bmatrix}=\begin{bmatrix} 0 & 0 & 0 & 1 & 4\\ 0 & 0 & 1 & 1 & 1\\ 0 & 0 & 0 & 6 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \\0 \\0 \\ 0\end{bmatrix} $
Obtain $\begin{bmatrix} 0 & 0 & 0 & 1 & 4\\ 0 & 0 & 1 & 1 & 1\\ 0 & 0 & 0 & 6 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \approx \begin{bmatrix} 0 & 0 & 1 & 1 & 1\\ 0 & 0 & 0 & 1 & 4\\ 0 & 0 & 0 & 0 & 24\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$
$\rightarrow dim(E_2)=3$
Hence, $J=\begin{bmatrix} 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$
