Suppose $A$ and $B$ are matrices, and that $B$ and $AB$ are both non-negative matrices (i.e. matrices with non-negative entries).
Can we infer any properties of $A$?
Obviously, if $A$ is itself non-negative then it is automatic that $AB$ is non-negative, but this is clearly not a requirement.
In the particular case I'm interested in, $A=e^{-Ct}$ for some $t\ge0$. Can one infer any properties of $C$, independent of $t$? (For example, if $A$ is non-negative, then $C$ must have non-positive off-diagonal elements, see e.g. Exponential of matrix with negative entries only on the diagonal, but it seems unlikely that this condition is a requirement.)
