The complex eigenvalue is confusing me. $\lambda_{1,2} = \frac{1}{2}(5 \pm i\sqrt{3})$
for $A=\begin{pmatrix}3&1\\-1&2\end{pmatrix}$
so I get this for $\lambda_1 = \frac{1}{2}(5 - i\sqrt{3})$ (I multiplied both equations by $2$):
\begin{equation} (1+i\sqrt{3})x+2y=0 \\ -2x - (1+i\sqrt{3})y = 0 \end{equation}
How do I solve for the eigenvector now? Thanks.
Edit
I don't think I fully understand the process of calculating eigenvectors \begin{equation} (1-i\sqrt{3})[(1+i\sqrt{3})x+2y=0]\\ -2x - (1+i\sqrt{3})y = 0 \end{equation}
\begin{equation} 4x+(2-2i\sqrt{3})y=0\\ -2x - (1+i\sqrt{3})y = 0 \end{equation}
This gives me $x=0$ and $y=0$. Does that mean there is no eigenvector?