I have trouble with how to find eigenvectors when you have a complex eigenvalue
For example the matrix
$$ \begin{pmatrix}
0 & 1\\
-2 & -2
\end{pmatrix}$$
Here you get the eigenvalues $-1$ and $ \pm i$
Where do i go from here to find a eigenvector. The solution says it should be the 2×1 matrix:
$$\begin{pmatrix}
1\pm i \\
-2
\end{pmatrix}$$
Best Answer
You do it just as you would for real eigenvalues.
For instance, if you want an eigenvector corresponding to the eigen value $-1+i$, you solve the system$$\left\{\begin{array}{ll}y=(-1+i)x\\-2x-2y=(-1+i)y,\end{array}\right.$$you will get that the solutions are the vectors of the form $\bigl((1+i)x,-2x\bigr)$. In particular, $(1+i,-2)$ is an eigenvector corresponding to the eigenvalue $-1+i$.