Find an eigenvector I am trying to find an eigenvector of the following matrix, where I get an eigenvector zero. But, to my knowledge, an eigenvector cannot be zero.
$$   \pmatrix{0&1\\0&i-1}\pmatrix{x\\y} = \pmatrix{0\\0}.$$

You are looking for an eigenvector $(x,y)$ associated to the eigenvalue $\lambda=0$.
$$
\begin{pmatrix}0&1\\0&i-1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = 0\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}
$$
On the other hand, 
$$\begin{pmatrix}0&1\\0&i-1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}y\\(i-1)y\end{pmatrix}$$
This implies $y=0$. So, an eigenvector is given by $\alpha\begin{pmatrix}1\\0\end{pmatrix}$ with $\alpha\in\mathbb{R}^*$.
