The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself.
Part (a) is quite easy. We get
$$\sigma_1 = 2\lambda, \mathbf{v}_1 =
\begin{pmatrix}
0 \\
0 \\
1
\end{pmatrix}
\sigma_2 = \lambda, \mathbf{v}_2 =
\begin{pmatrix}
1/\sqrt{2} \\
1/\sqrt{2} \\
0
\end{pmatrix}
\sigma_3 = -\lambda, \mathbf{v}_3 =
\begin{pmatrix}
1/\sqrt{2} \\
-1/\sqrt{2} \\
0
\end{pmatrix}
$$
There are two ways...