Answer
See below
Work Step by Step
1. Find eigenvalues:
(A-$\lambda$I)$\vec{V}$=$\vec{0}$
$\begin{bmatrix} -1-\lambda & -1 & 0\\ 0 & -1-\lambda & -2\\ 0 & 0 & -1-\lambda \end{bmatrix}\begin{bmatrix} v_1\\ v_2 \\ v_3 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \\0 \end{bmatrix}$
$\begin{bmatrix} -1-\lambda & -1 & 0\\ 0 & -1-\lambda & -2\\ 0 & 0 & -1-\lambda \end{bmatrix}=0$
$( \lambda+1)^3=0$
$\lambda_1=\lambda_2=\lambda_3=-1$
2. Find eigenvectors:
For $\lambda=-1$
let $B=A-\lambda_1I$
$B=\begin{bmatrix} -1-\lambda & -1 & 0\\ 0 & -1-\lambda & -2\\ 0 & 0 & -1-\lambda \end{bmatrix}=\begin{bmatrix} 0 & -1 & 0\\ 0 & 0 & -2\\ 0 & 0 & 0 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \\0 \end{bmatrix} $
Let $r,s,t$ be free variables.
$\vec{V}=r(2,0,0)+s(0,-2,0)+t(2,0,0)\\
E_1=\{(2,0,0);(0,-2,0);(2,0,0)\} \\
\rightarrow dim(E_2)=3$
Hence, $S=\begin{bmatrix} 2& 0 & 0 \\ 0 & -2& 0 \\
0 & 0 & 1 \end{bmatrix}$
