Q:

Eigenvectors of a zero diagonal matrix

Question: Find a unitary matrix $P$ such that $P^{-1}AP$ is diagonal, where 
$$A=\
\left( \begin{matrix}
  0 & 2i & i & 0 \\
  2i & 0 & 0 & i \\
  i & 0 & 0& 2i \\
0 & i & 2i & 0
\end{matrix}\right) 
$$
and then calculate $A^4$.
What I know: I know how to find a unitary matrix, but I'm not sure how to do it quickly in this case. I require the eigenvectors of this matrix. I would be able to do this but it would take quite a while for a 4x4 matrix. However it is symmetric and it has zeros on the diagonal, so I feel there is a way to spot them.

A:

Note that you can obtain the matrix
$$\begin{pmatrix}2i&0&0&i\\0&2i&i&0\\0&i&2i&0\\i&0&0&2i \end{pmatrix}$$
by permuting rows.
Also, change the fourth row $R_4$ with $R_4-\frac12R_1$, and $R_3$ with $R_3-\frac12R_2$:
$$\begin{pmatrix}2i&0&0&i\\0&2i&i&0\\0&0&\frac32i&0\\0&0&0&\frac32i \end{pmatrix}$$

