Here is a question which arises from physics.

**Let $A$, $B$ be two symmetric real-valued matrices. What conditions should the matrices meet to make $AB$ has a pure complex eigenvalue ($Im(\lambda) \neq 0$)?**

Although I'm interested in complex eigenvalues it might be useful to see when $AB$ has real spectrum.

For instance, if $A$ is a positive form, then the spectrum of $AB$ is real-valued. Or if $[A,B] = 0$ then again spectrum of product is real. So there is no way to find a purely complex eigenvalue and we are done. What will happen if $A$ isn't positive definite? I'm searching for useful criteria that doesn't require positivity. Or maybe there exist criteria which ensure that $AB$ must have a purely complex eigenvalue.

Has anybody heard about this? In what direction should I look?

**UPD** For instance, if matrices are 2x2, the criteria is the following:

*$AB$ has only real eigenvalues $\Leftrightarrow$ $\exists$ a linear combination $\lambda A + \mu B^{-1}$ with real ${\lambda, \mu}$, s.t. it is positively defined*

anyreal matrix can be written as the product of two real symmetric matrices -- this makes things "easy" speaking generically. $\endgroup$3more comments