This question has popped up at me several times in my research in differential equations and other areas:
Let $A$ be a real $N \times N$ matrix. How are the eigenvalues of $A$ and $A + A^T$ related? Of course, if $A$ is symmetric, the answer is easy: they are the same up to factor or $2$, since then $A + A^T = 2A$. But if $A \ne A^T$?
I'm particularly interested in the question of the real parts of the eigenvalues. How are the real parts of the eigenvalues of $A$ related to the (necessarily) real eigenvalues of $A + A^T$?
Answers for complex matrices appreciated as well.
Any references, citings, or explanations at any level of detail will be appreciated.
Thanks in advance.