So $A$ satisfies these constrains, $A^{-1}$ must not satisfies. The question is not reasonable. If matrices $A$ and $B$ are given, does there always exists $k \in Z^+$ such that $C=A(kI + B)$ has the property 2 (i.e. the max eigenvalue of $C$ is real and has multiplicity of 1)? If there always exist $k$, how can we find it? Thanks, David

Thank you Igor. Can we add more constrains (properties) on $A$ and $B$ such that the max eigenvalue of $AB$ is real and has multiplicity of 1? If so, can you suggest some kind of constrains that I can use?