Let Y be an n × p multivariate normal random matrix with general covariance ΣYand 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.
Hu, J. (2010). Equivalent conditions for noncentral generalized Laplacianness and independence of matrix quadratic forms. Linear Algebra and Its Applications, 433(4), 796–809. https://doi.org/10.1016/j.laa.2010.04.010