Find a basis $\gamma$ with respect to which both of the following lienar transformations on $\mathbb{R^3}$ become diagionalised (the matrices below are the matrices with respect to the standard basis):
$$S=\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} $$
and $$T=\begin{bmatrix} 2 & -1 & 2 \\ -1 & 2 & 2 \\ 2 & 2 & -1 \\ \end{bmatrix}$$
I found a basis of eigenvectors for S and another one for T but they are not the same, how should I proceed?
Hint: Try the transformation matrix (or it's inverse depending on what you call the transformation matrix) $$\begin{bmatrix} 1 & 1 & -2\\ 1 & -1 &-2\\ -2& 0 &-2\\ \end{bmatrix}$$