Solution
= Solution
The <Euclidean norms> are $\|v_\pm\|=2$ and $\|v_0\|=\sqrt2$. With normalized <eigenvectors> as rows,
$$
\boxed{R=\begin{pmatrix}1/2&1/\sqrt2&1/2\\1/\sqrt2&0&-1/\sqrt2\\1/2&-1/\sqrt2&1/2\end{pmatrix},\qquad RMR^T=\operatorname{diag}(\sqrt2,0,-\sqrt2).}
$$
The rows form an <orthonormal basis>, so $RR^T=I$ and $R$ is an <orthogonal matrix>. The zero <eigenvalue> means $Mv_0=0$ for a nonzero <vector>; thus \b[$M$ is singular] and has no <matrix inverse>.