If $A$ and $B$ are positive-definite matrices, is $AB$ positive-definite? I've managed to prove that if $A$ and $B$ are positive definite then $AB$ has only positive eigenvalues. To prove $AB$ is positive definite, I also need to prove $(AB)^\ast = AB$ (so $AB$ is Hermitian). Is this statement true? If not, does anyone have a counterexample?
Thanks,
Josh

In general no, because for Hermitian $A$ and $B$, $(AB)^* = AB$ if and only if $A$ and $B$ commute.  On the other hand, $ABA$ and $BAB$ can be proven to be positive definite.
