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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.

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    What have you tried so far? To me there doesn't seem to be anything special when you do that. Except perhaps whether $i$ is the imaginary unit or just a real constant. – 2017-01-02
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    @skyking I've seen the solution and in it the eigenvalues and eigenvectors are just stated without any reasoning. – 2017-01-02
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    Haven't you done any attempts at trying to determine the eigenvalues or eigenvectors? Do you know how one usually determines them? What knowledge of eigenvectors do you have? For one to formulate a useful answer one need to know which level to put it on (ie which level you're on). – 2017-01-02
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    @skyking I know how to find them usually, and I know how to find them for a 4x4 matrix, but I am trying to understand what property the lecturer used in order to find them quickly for a matrix which has these properties. – 2017-01-02

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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 $R_3-\frac12R_2$: $$\begin{pmatrix}2i&0&0&i\\0&2i&i&0\\0&0&\frac32i&0\\0&0&0&\frac32i \end{pmatrix}$$