Let:
$A=\begin{pmatrix} 1&i&i\\ i&1&i\\ i&i&1 \end{pmatrix}$
I have proved that $A$ is normal. Now I want to find a unitary matrix $P$ such that $P^*AP$ is a diagonal matrix ($P^*$ is the conjugate transpose of $P$).
The eigenvalues of $A$ are: $\{1+2i, 1-i\}$ and the eigenvectors: $\{(1,1,1)^t, (-1,0,1)^t,(-1,1,0)^t\}$
How can I get this matrix $P$?