Say I have two matrices $A$ and $B$. I know that they each have real eigenvalues. Clearly if they're Hermitian, then their sum would also have real eigenvalues, since then $A+B$ is Hermitian. However, I'm wondering what happens if they are not necessarily Hermitian. Can we say anything about $A+B$? Are the eigenvalues of $A+B$ real? I have tried to come up with a counterexample but didn't manage and didn't get very far manipulating the characteristic equation. What if just one of them is Hermitian?
Every real square matrix $C$ is the sum of two real matrices $A$ and $B$ with real spectra, where $A$ can be chosen to be symmetric. In fact, let $H$ and $K$ be the symmetric and skew-symmetric parts of $C$, and let $L$ be the lower triangular part of $K$. Then one can take $A=H+L+L^T$ and $B=-2L^T$.
But obviously, not every real square matrix $C$ has a real spectrum.