Answer
See below
Work Step by Step
1. Find eigenvalues:
(A-$\lambda$I)$\vec{V}$=$\vec{0}$
$\begin{bmatrix} 1-\lambda & 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 1-\lambda & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1-\lambda & 1 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 1-\lambda & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1-\lambda & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 &1-\lambda & 0 & 0\\ 0 & 0 & 0 & 0 &0 & 0 & 1-\lambda & 1 \\ 0 & 0 & 0 & 0 & 0 & 0&0 &1-\lambda\end{bmatrix}\begin{bmatrix} v_1\\ v_2 \\ v_3 \\ v_4 \\ v_5 \\ v_6 \\ v_7 \\ v_8 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \\0 \\0 \\ 0 \\0 \\ 0\\ 0 \end{bmatrix}$
$\begin{bmatrix} 1-\lambda & 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 1-\lambda & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1-\lambda & 1 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 1-\lambda & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1-\lambda & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 &1-\lambda & 0 & 0\\ 0 & 0 & 0 & 0 &0 & 0 & 1-\lambda & 1 \\ 0 & 0 & 0 & 0 & 0 & 0&0 &1-\lambda\end{bmatrix}=0$
$(\lambda-1)^8=0$
$\lambda_1=1$ with multiplicity 8.
Fỏ $\lambda=1$
$\begin{bmatrix} 1-\lambda & 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 1-\lambda & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1-\lambda & 1 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 1-\lambda & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1-\lambda & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 &1-\lambda & 0 & 0\\ 0 & 0 & 0 & 0 &0 & 0 & 1-\lambda & 1 \\ 0 & 0 & 0 & 0 & 0 & 0&0 &1-\lambda\end{bmatrix} \approx \begin{bmatrix} 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 1-\lambda & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 &0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0&0 & 0\end{bmatrix}$
$\rightarrow \dim =4$
Hence, $J=\begin{bmatrix} 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 &1 & 0 & 0\\ 0 & 0 & 0 & 0 &0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0&0 &1 \end{bmatrix}$
