Suppose that $A$ and $B$ are both Hermitian and positive definite, and they are both $n\times n$ matrices, moreover, $AB=BA$. It is easy to prove that $AB$ is Hermite. Is $AB$ positive definite?
If your answer is that $AB$ is positive definite，then how to prove this conclusion?
If your answer is that $AB$ is not positive definite, please make an example.
Thank you!
The matrices are diagonalisable, and commute so they are simultaneously diagonalisable. So each eigenvalue of $AB$ is a product of an eigenvalue of $A$ and an eigenvalue of $B$. Both $A$ and $B$ have all-positive eigenvalues...