I am trying to find the eigenvectors for a matrix $A$ with $\lambda_{1,2} = -1 \pm i\nu$ and I computed
\begin{equation} A-\lambda_{1} I = \begin{bmatrix} 2-i\nu & \mu+\nu& \mu & \nu+2\\ 0 & \nu-i\nu & 0 & 2\nu\\ -\mu & 2-\nu & 2-i\nu& -\mu-2\nu\\ 0 & -\nu& 0 & -\nu- i \nu \end{bmatrix} \end{equation}
Now it is given that \begin{equation} \begin{bmatrix} 0\\ -1\\ 0\\ 1 \end{bmatrix} + i \begin{bmatrix} 0\\ -1\\ 0\\ 0 \end{bmatrix} \end{equation} is solution for row 2 and 4. I really do not see why. And also, how would I continue from here to find the corresponding eigenvectors?