Problem

$$M=\begin{pmatrix} 0&-1&3\\ -1&4&-1\\ 3&-1&0 \end{pmatrix}$$

Given that $\begin{pmatrix}1\\-2\\1\end{pmatrix}$ is an eigenvector of $M$

(a) determine its corresponding eigenvalue.

Given that $-3$ is an eigenvalue of $M$

(b) determine a corresponding eigenvector.

Hence, given that $\begin{pmatrix}1\\1\\1\end{pmatrix}$ is also an eigenvector of $M$

(c) determine a diagonal matrix $D$ and an orthogonal matrix $P$ such that $D=P^TMP$