Is there anything that says when covariance matrices commute? For one, I did some digging around the internet and couldn't find much talk about when symmetry connects to commutativity. (This would be nice, because covariance matrices of stationary processes are symmetric.) I couldn't find anything at all about the commutativity of covariance matrices in general, either.
I know that symmetry depends on choice of basis, but what if both matrices are symmetric in the initial basis?