Do complex eigenvalues for planar systems with a real valued matrix always have complex-valued eigenvectors?
Solution 1:
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.