Wishartness and independence of matrix quadratic forms in a normal random matrix

Let Y be an n×p multivariate normal random matrix with general covariance ΣY. The general covariance ΣY of Y means that the collection of all np elements in Y has an arbitrary covariance matrix. A set of general, succinct and verifiable necessary and sufficient conditions is established for matrix quadratic forms Y′WiY's with the symmetric Wi's to be an independent family of random matrices distributed as Wishart distributions. Moreover, a set of general necessary and sufficient conditions is obtained for matrix quadratic forms Y′WiY's to be an independent family of random matrices distributed as noncentral Wishart distributions. Some usual versions of Cochran's theorem are presented as the special cases of these results.