Consider two symmetric and real matrices $A,B\in\mathbb{R}^n$ and definie $A+iB$. Note that $A+iB$ is not hermitian in this case. There are many results based on Brendixson and Courant-Fischer, saying, that for every eigenvalue $\lambda+i\mu$ of $A+iB$ and $\beta_{min},\beta_{max}$ the minimal and maximal eigenvalue of $B$, we get that $\beta_{min}\leq \mu\leq \beta_{max}$.
My question now is: Are there any reversal inequalities? What can we say about the eigenvalues of $B$ knowing those of $A+iB$?. The case in which I am interested the most is the one with $\mu>0$ for all eigenvalues $\lambda+i\mu$ of $A+iB$.
Do you have any idea how to do this? Thanks for your help!
