Let Y be an n × p multivariate normal random matrix with general covariance ΣY and W be a symmetric matrix. In the present article, the property that a matrix quadratic form Y′ WY is distributed as a difference of two independent (noncentral) Wishart random matrices is called the (noncentral) generalized Laplacianness (GL). Then a set of algebraic results are obtained which will give the necessary and sufficient conditions for the (noncentral) GL of a matrix quadratic form. Further, two extensions of Cochran's theorem concerning the (noncentral) GL and independence of a family of matrix quadratic forms are developed. © 2010 Elsevier Inc. All rights reserved.
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