8.
$$\mathbf { A } = \left( \begin{array} { l l l }
1 & 0 & 4
0 & 5 & 4
4 & 4 & 3
\end{array} \right)$$
- Verify that \(\left( \begin{array} { r } 2
- 2
1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\) and find the corresponding eigenvalue. - Show that 9 is another eigenvalue of \(\mathbf { A }\) and find the corresponding eigenvector.
- Given that the third eigenvector of \(\mathbf { A }\) is \(\left( \begin{array} { r } 2
1
- 2 \end{array} \right)\), write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that
$$\mathbf { P } ^ { \mathrm { T } } \mathbf { A } \mathbf { P } = \mathbf { D } .$$