connections between eigenvalues of $B$ and $A+iB$

Question

I am interested in connections between the eigenvalues of $B$ and $A+iB$ with $A,B$ symmetric and real. 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 $\beta_{min}\leq \mu\leq \beta_{max}$.

But I didnt find anything about a reversal inequation: I would like to say something about the eigenvalues of $B$ when knowing those of $A+iB$. Especially for the case when we have $\mu>0$ for all eigenvalues $\lambda+i\mu$ of $A+iB$.

Do you have any idea how to do this? Thanks for your help!

Answer 1

• You could look at the field of values of $A + iB$ (see Properties of the field of values).
• Note that, for any vector $z$, we have $$ \mathrm{Im}\left( z^* A z + i z^* B z \right) = z^* B z $$ and when $z$ is an eigenvector for the eigenvalue $\lambda + i \mu$, we write $$ \left( A z + i B z \right) = \left( \lambda + i \mu \right) z \; \Rightarrow \; z^* B z = \mu z^* z $$