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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?

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To avoid base dependence, you might want to ask after "symmetrizable matrices" instead of symmetric ones. –  A Walker May 27 '13 at 6:27
Covariance matrices are not really matrices in the sense that they don't really define linear operators (they define bilinear forms), so multiplying them isn't a natural thing to do. –  Qiaochu Yuan May 27 '13 at 6:37
@Qiaochu You're right; the only reason why I need to know is in order to completely jointly diagonalize them. –  Trevor Alexander May 27 '13 at 7:29

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