Answer
See below
Work Step by Step
1. Find eigenvalues:
(A-$\lambda$I)$\vec{V}$=$\vec{0}$
$\begin{bmatrix} -\lambda & 1 & 1 & 1 & 1 \\ 1 & -\lambda & 1 & 1 & 1\\ 1& 1 & -\lambda & 1 & 1 \\ 1 & 1 & 1 & -\lambda & 1\\ 1 & 1 & 1 & 1 & -\lambda\end{bmatrix}\begin{bmatrix} v_1\\ v_2 \\ v_3 \\ v_4 \\ v_5 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \\ 0 \\ 0 \\ 0\end{bmatrix}$
$\begin{bmatrix} -\lambda & 1 & 1 & 1 & 1 \\ 1 & -\lambda & 1 & 1 & 1\\ 1& 1 & -\lambda & 1 & 1 \\ 1 & 1 & 1 & -\lambda & 1\\ 1 & 1 & 1 & 1 & -\lambda\end{bmatrix}=0$
$\lambda_1=\lambda_2=\lambda_3=\lambda_4=-1$
2. Find eigenvectors:
For $\lambda=-1$
let $B=A-\lambda_1I$
$B=\begin{bmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1\\ 1& 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1
& 1\end{bmatrix}=\begin{bmatrix} 0\\ 0 \\ 0 \\ 0 \\ 0\end{bmatrix} $
Let $r,s,t,v$ be free variables.
$\vec{V}=r(-1,0,0,1,0)+s(-1,0,1,0,0)+t(-1,0,0,0,1)+v(-1,1,0,0,0)\\
E_1=\{(-1,0,0,1,0);(-1,0,1,0,0);(-1,0,0,0,1);(-1,1,0,0,0)\} \\
\rightarrow dim(E_2)=4$
