I have to find the eigenvalues and eigenvectors for this matrix:

$\displaystyle \begin{pmatrix}4&2i\\2i&6\end{pmatrix}$

I found eigenvalues of $\displaystyle \lambda=5\pm\sqrt{3}i$

I'm having trouble finding the eigenvectors

$\displaystyle ker\begin{pmatrix}-1\pm\sqrt{3}i&2i\\2i&1\pm\sqrt{3}i\end{pmatrix}$

I know how to find eigenvectors, but this one seems confusing because of the complex eigenvalues