[Math] If two covariance matrices commute, is their product a covariance matrix

Question

Let $A$ and $B$ be two covariance matrices such that $AB=BA$. Is $AB$ a covariance matrix?

A covariance matrix must be symmetric and positive semi definite. The symmetry of $AB$ can be proved as follows:
$$(AB)^T = B^TA^T = BA = AB$$

The question is, how to prove or disprove the positive semi definitive character of $AB$?

Answer 1

Two commuting matrices can be diagonalized by the same matrix. The positive semi definite follows immediately.