Let $A$ be real and let $\lambda = \alpha + i \beta$ be a complex eigenvalue of $A$ with eigenvector $x + iy$, show that the space spanned by $x$ and $y$ is an invariant subspace of $A$.
What I believe I need to show: I think I want to show $Av=xv$ and $Av=yv$ where $v$ is the eigenvector given above. Is this assumption correct? And if so, I wasn't having any luck proving this. If this isn't what I'm suppose to prove, could someone explain what I am suppose to try and prove for this problem. Thank you.