Upon solving an eigenvector problem, I've been summoned to find the kernel of the following matrix:

$\displaystyle \[

\left( {\begin{array}{cc}

-1-\sqrt{3}i & 2i \\

2i & 1-\sqrt{3}i \\

\end{array} } \right)

\]$

Perhaps I'm just sleep deprived, but I'm having trouble getting this into row-echelon form. Any advice?