$\displaystyle A=\begin{bmatrix}1&2&2\\2&1&-2\\2&-2&1\end{bmatrix}$

I have found the eigenvalues and vectors to be $\displaystyle \lambda=3,3,-3$ and eigenvectors $\displaystyle \left\{\begin{bmatrix}1\\1\\0\end{bmatrix},\begin{ bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}-1\\1\\1\end{bmatrix}\right\}$.

Now I need to find an orthogonal matrix such that $\displaystyle P^tAP=D$.

D being my diagonal matrix.

I tried Gram-Schmidting my eigenspace but that didn't work. What do I need to do?