Answer
See solution
Work Step by Step
Let $u_1=\begin{bmatrix}
-1/\sqrt{2}\\1/\sqrt{2}\\0
\end{bmatrix}$, $u_2=\begin{bmatrix}
-1/\sqrt{6}\\-1/\sqrt{6}\\2/\sqrt{6}
\end{bmatrix}$, and $u_3=\begin{bmatrix}
1/\sqrt{3}\\1/\sqrt{3}\\1/\sqrt{3}
\end{bmatrix}$.
The the spectral decomposition of A is $8u_1u_1^T+6u_2u_2^T+3u_3u_3^T$
$=8\begin{bmatrix}
1/2&-1/2&0\\-1/2&1/2&0\\0&0&0
\end{bmatrix}+6\begin{bmatrix}
1/6&1/6&-2/6\\1/6&1/6&-2/6\\-2/6&-2/6&4/6
\end{bmatrix}+3\begin{bmatrix}
1/3&1/3&1/3\\1/3&1/3&1/3\\1/3&1/3&1/3\\
\end{bmatrix}$
