Answer
See below
Work Step by Step
1. Find eigenvalues:
(A-$\lambda$I)$\vec{V}$=$\vec{0}$
$\begin{bmatrix}
3-\lambda & -2 & 3 & -2\\
-2 & 3-\lambda & -2 &3 \\
3& -2 & 3-\lambda & -2\\
-3 & 3 & -2 & 3-\lambda
\end{bmatrix}\begin{bmatrix}
v_1\\
v_2 \\
v_3\\
v_4
\end{bmatrix}=\begin{bmatrix}
0\\
0 \\
0\\
0
\end{bmatrix}$
$\begin{bmatrix}
3-\lambda & -2 & 3 & -2\\
-2 & 3-\lambda & -2 &3 \\
3& -2 & 3-\lambda & -2\\
-2 & 3 & -2 & 3-\lambda
\end{bmatrix}=0$
$\lambda_1=0, \lambda_2=2, \lambda_3=2, \lambda_4=10$
2. Find eigenvectors:
For $\lambda=0$
$E_1=\{(0,1,0,-1)+(1,0,-1,0)\}
\rightarrow dim(E_1)=2$
The eigenvectors span $\{(0,1,0,-1),(1,0,-1,0)\}$ in $R$
For $\lambda=2$
$E_2=\{(1,1,1,1)\}
\rightarrow dim(E_2)=1$
The eigenvectors span $\{(1,1,1,1)\}$ in $R$
For $\lambda=10$
$E_3=\{(-1,1,-1,1)\}
\rightarrow dim(E_3)=1$
The eigenvectors span $\{(-1,1,-1,1)\}$ in $R$
We obtain $S=\begin{bmatrix}
0 & 1 & 1 & -1\\
1 & 0 & 1 & 1 \\
0 & -1 & 1 & -1\\
-1 & 0 & 1 & 1
\end{bmatrix} \rightarrow
S^{-1}AS=diag(0,2,1,0)$
