Suppose $A,B$ are symmetric and positive definite (real) matrices such that $AB$ is not symmetric (equivalently $AB\neq BA$). My question is, what are sufficient and/or necessary conditions for $AB$ being positive definite in this case?
For clarification, I call a square matrix $C$'positive definite' if $x^TCx>0$ for any vector $x$, regardless of whether $C$ is symmetric or not.
