4.
$$\mathbf { M } = \left( \begin{array} { r r r }
0 & - 1 & 3
- 1 & 4 & - 1
3 & - 1 & 0
\end{array} \right)$$
Given that \(\left( \begin{array} { r } 1
- 2
1 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\)
- determine its corresponding eigenvalue.
Given that - 3 is an eigenvalue of \(\mathbf { M }\)
- determine a corresponding eigenvector.
Hence, given that \(\left( \begin{array} { l } 1
1
1 \end{array} \right)\) is also an eigenvector of \(\mathbf { M }\) - determine a diagonal matrix \(\mathbf { D }\) and an orthogonal matrix \(\mathbf { P }\) such that \(\mathbf { D } = \mathbf { P } ^ { \mathrm { T } } \mathbf { M P }\)