As I posted here and here I'm studying Jordan forms and similar concepts.
I've got a problem with complex eigenvalues in jordan real matrices. I know (at least I think so) how to compute the Jordan blocks for real eigenvalues. However, I don't know how to do it with complex ones. Let's illustrate this with an example:
Let $M$ be
$$\begin{bmatrix}
2 & 2 & 4 & 6 & 4\\
0 & 0& -4 & -6 &-4\\
0 & 0 & 2 & 4 & 4\\
0 & 2 & 0 & -4 & -4\\
-2 & -4 & -2 & 2 & 2\\ \end{bmatrix}$$
The charasteristic polynomial is $(x-2)(x^2+4)^2$. I know the Jordan real form is: $$\begin{bmatrix} 2 & 0 & 0 & 0 & 0\\ 0 & 0& -2 & 0 &0\\ 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2\\ 0 & 0& 0 & 2 & 0\\ \end{bmatrix}$$
but I don't understand why. The block corresponding to the eigenvalue $2$ is clear, but I'm completely lost about the complex eigenvalues and how to arrive to that matrix.
Thanks in advance.