6. The symmetric matrix \(\mathbf { M }\) has eigenvectors \(\left( \begin{array} { l } 2
2
1 \end{array} \right) , \left( \begin{array} { r } - 2
1
2 \end{array} \right)\) and \(\left( \begin{array} { r } 1
- 2
2 \end{array} \right)\) with eigenvalues 5, 2 and - 1 respectively.

1. Find an orthogonal matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { \mathrm { T } } \mathbf { M } \mathbf { P } = \mathbf { D }$$ Given that \(\mathbf { P } ^ { - 1 } = \mathbf { P } ^ { \mathrm { T } }\)
2. show that $$\mathbf { M } = \mathbf { P D P } ^ { - 1 }$$
3. Hence find the matrix \(\mathbf { M }\).