On the Distribution of Matrix Quadratic Forms

Author: Singull Martin  

Publisher: Taylor & Francis Ltd

ISSN: 0361-0926

Source: Communications in Statistics: Theory and Methods, Vol.41, Iss.18, 2012-09, pp. : 3403-3415

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Abstract

A characterization of the distribution of the multivariate quadratic form given by X A X′, where X is a p × n normally distributed matrix and A is an n × n symmetric real matrix, is presented. We show that the distribution of the quadratic form is the same as the distribution of a weighted sum of non central Wishart distributed matrices. This is applied to derive the distribution of the sample covariance between the rows of X when the expectation is the same for every column and is estimated with the regular mean.