Is there any relation between the eigenvalues of $A+A^\star$ and the eigenvalues of $A$ or $A^\star$ (assuming they are not Hermitian)?

Question

Here $A^\star$ denotes conjugate transpose.

Answer 1

$A+A^*$ is an hermitian matrix that is 2 times the hermitian part of the matrix $A$. $$ \Re(A) := \frac{A+A^*}{2} $$

This means that we can apply the following theorem

Let $I=[\lambda_1,\lambda_n]$ be the real interval between the minimum eigenvalue of $\Re(A)$ and the maximum. If $\mu_1,\mu_2,\dots,\mu_n$ are the eigenvalues of $A$, then their real parts lie inside $I$.

to prove that the real part of any eigenvalue $\mu$ of $A$ is $$\frac{\nu_1}2\le\mu\le \frac{\nu_n}2$$ where $\nu_1$ and $\nu_n$ are the smallest and biggest eigenvalues of $A+A^*$.