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 Σ Y . The general covariance Σ Y of Y means that the collection of all np elements in Y has an arbitrary np × np covariance matrix. A set of general, succinct and verifiable necessary and sufficient conditions is established for matrix quadratic forms Y ′ W i Y ʹs with the symmetric W i ʹs 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 ′ W i Y ʹs to be an independent family of random matrices distributed as noncentral Wishart distributions. Some usual versions of Cochranʹs theorem are presented as the special cases of these results.