3.
$$\mathbf { M } = \left( \begin{array} { r r r }
- 2 & 5 & 0
5 & 1 & - 3
0 & - 3 & 6
\end{array} \right)$$
Given that \(\mathbf { i } + \mathbf { j } + \mathbf { k }\) is an eigenvector of \(\mathbf { M }\),
- determine the corresponding eigenvalue.
Given that 8 is an eigenvalue of \(\mathbf { M }\),
- determine a corresponding eigenvector.
- Determine a diagonal matrix \(\mathbf { D }\) and an orthogonal matrix \(\mathbf { P }\) such that
$$\mathbf { D } = \mathbf { P } ^ { \mathrm { T } } \mathbf { M P }$$