Suppose we have two covariance matrices $A$ and $B$. They satisfy the condition $AB=BA$. Is $AB$ a covariance matrix?
We can easily check that $(AB)'=B'A'=BA$, then $AB$ is symmetric. But I have no idea how to check it is positive semi-definite. I can't come up with an example showing it isn't a covariance matrix, either.
Any help would be appreciated.