I know that the covariance matrix of a random vector must be symmetric and positive semidefinite. What about a covariance matrix between two random vectors? Does it have the same properties?
2026-03-30 14:58:54.1774882734
Properties of covariance matrix of two random vectors
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No, it doesn’t. For instance, from the facts that you cite it follows that the cross-covariance matrix of $X$ and $-X$ does not have these properties (where $X$ is a random vector that is not almost surely zero).