How to compute the Real Jordan Normal Form of a specific matrix I am getting stuck in a homework problem and am trying to figure out how to compute the real jordan normal form of a specific matrix $$A=\begin{bmatrix}-1&-1&0 \\
2&1&0\\
1&2&1
\end{bmatrix}$$
In particular I am interested in what the transformation matrices $P$ and $P^{-1}$ would need to be. When plugging into wolfram it gives me a Jordan form of $$J=\begin{bmatrix}i&0&0\\
0&-i&0\\
0&0&1\\
\end{bmatrix}$$
This is not what I am looking for. I need the entries to be real to help compute the center manifold for a system. Can someone please help me out.

I don't know  about dynamical systems, but I can help with the Real Jordan Form.
It is: $J_{\mathbb R}=
\begin{bmatrix} 
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 1
\end{bmatrix}=
\begin{bmatrix} 
Re(\mu) & Imm(\mu) & 0 \\
-Imm(\mu) & Re(\mu) & 0 \\
0 & 0 & 1
\end{bmatrix}
$
where $\mu=i$ is the complex eigenvalue.
You can see that the block for $\mu$ and the one for $\bar\mu$ explode in one double size block.
Don't esitate to ask for explanations.
