How to quickly compute the Jordan Matrix for this $4\times 4$ matrix

Question

Now I've learnt how to compute the Jordan matrix of a $2\times 2$ block. I wonder how you can do this fast without numerical tools.

Let my matrix be:

$A = \begin{bmatrix} 0 & −1 &1 & 0\\ 1 & 0 &0 & 1\\ 0 & 0 &0 &−1\\ 0 & 0 &1 & 0 \end{bmatrix} $

Does anyone know how to quickly compute the Jordan Block Matrix for this matrix $A$ ?

Answer 1

Expand $\det(\lambda I - A) = 0$ to get $$L^4 + 2L^2 + 1 = 0$$ which is a biquadratic which can be quickly solved to get the roots $[i, i, -i, -i]$, so the Jordan form is:

$$\begin{bmatrix} i & 1 & 0 & 0 \\ 0 & i & 0 & 0 \\ 0 & 0 & -i & 1 \\ 0 & 0 & 0 & -i \\ \end{bmatrix}$$