Diagonalization and invariant subspaces of a symmetric $3\times 3$ matrix

A linear transformation A writes in some basis in \(\mathbb { R } ^ { n }\) as follows, \(A = { \left[ \begin{array} { l l l } { 0 } & { 2 } & { 1 } \\ { 2 } & { 8 } & { 2 } \\ { 1 } & { 2 } & { 0 } \end{array} \right] }\)

1. Write the transition matrix to the basis composed of the eigenvectors of A.

2. Write the transformation A in this basis.

3. Determine the invariant subspaces of A.