I am trying to compute the eigenvector associated to $\lambda_{1}=i$, which (for my matrix) means I have to solve the system of four equations:
$$\begin{cases}(\xi-i)x-y=0 \\ x+(\xi-i)y=0 \\ (\xi-i)z-w=0 \\ 4z+(\xi-i)w=0\end{cases}$$
$\implies \begin{cases}x=-y \\4z=-w\end{cases}$
But these two equations are linearly independent, so how do I write my eigenvector?
Considering your last set of equations, the rank of matrix corresponding to your homogeneous system of 4 variables is 2 thereby giving 4-2=2 linearly independent non-zero eigenvectors.
The two eigenvectors are of type [a -a 0 0] and [0 0 -b 4b] where a ,b∈C