Let $A$ be a real $4 \times 4$ matrix. Supose $i,-i$ are the eigenvalues of $A$. Show that there exists an invertible matrix $P$ such that $PAP^{-1}$ is either
$$\begin{pmatrix} 0&-1&0&0\\ 1&0&0&0\\ 0&1&0&-1\\ 0&0&1&0 \end{pmatrix}, \begin{pmatrix} 0&-1&0&0\\ 1&0&0&0\\ 0&0&0&-1\\ 0&0&1&0 \end{pmatrix}.$$
My professor gives me some hints: The characteristic polynomial of $A$ is $f_A (x)=(x^2 +1)^2$ . Therefore, by structure theorem of modules over a PID, $Im(A)$ is isomorphic to $\mathbb{R}[x]/((x^2 +1)^2 )$ or $\mathbb{R}[x]/(x^2 +1) \times \mathbb{R}[x]/(x^2 +1)$ ...
