My question: suppose $A$ and $B$ are two Hermitian positive-definite matrices (they don't commute: $AB\neq BA$). Are all eigenvalues of $AB$ positive?
I am completely confused. I know because $A$ and $B$ don't commute, $AB$ is not positive-definite. But according to: https://en.wikipedia.org/wiki/Positive-definite_matrix having all positive eigenvalues is equivalant to being positive-definite. So, since $AB$ is not positive-definite, not all its eignevalues are positive.
But here: For real matrices, if $A$ and $B$ are both positive-definite, show that all of $AB$'s eigenvalues are positive. in the answer it is somehow shown that all eigenvalues of $AB$ are positive.
So, what is the difference between these two cases? and what is the correct answer to my question?