# Do complex eigenvalues for planar systems with a real valued matrix always have complex-valued eigenvectors?

sorry if this is a dumb question, but I was doing some homework and I noticed that whenever I solved a planar system which had complex eigenvalues, I would always end up with complex eigenvectors. I was wondering whether I could ever somehow get a totally real-valued eigenvector from a complex eigenvalue.

The short answer is no. Let's suppose that $$A$$ is a real matrix with a complex eigenvalue $$\lambda$$ and an associated (nonzero) real eigenvector $$v$$. Then by the definition of eigenvector, $$Av = \lambda v$$. So $$\lambda v$$ must be complex since $$v$$ has real entries but $$\lambda$$ is not real; on the other hand, $$Av$$ must be real since both $$A$$ and $$v$$ have real entries by assumption.
This is impossible. Now we can say that if $$A$$ is a real matrix with a complex eigenvalue, then any associated eigenvector cannot have only real entries.