Eigenvalues of $A+A^{T}$

Question

Knowing the eigenvalues of $A$, what we can say about the eigenvalues of $A+A^{T}$?

I know that $A$ and $A^T$ have the same spectrum, but also I think that it is not true that $\lambda_{i,(A+A^T)} = 2\lambda_{i,A}$ where $\lambda_{i,A}$ is the $i$-th eigenvalue of $A$. For example if $A$ is a orthogonal matrix we have: $A+A^T=A+A^{-1}$ and thus $\lambda_{i,(A+A^T)} = \lambda_{i,A} + \frac{1}{\lambda_{i,A}}$.

Answer 1

On this forum post on MathOverflow the user Denis Serre gave some interesting insights. Probably is the most you can say about your problem

Answer 2

$A^{T}$ can be equal to $-A$ so there is hardly anything you can say about eigen values of $A+A^{T}$.