Applications of a Certain Representation of the Wishart Matrix

Abstract

Apart from pre- and post-multiplication by a fixed matrix and its transpose, the Wishart matrix $\mathbf{A}$ can be written as the product of a triangular matrix and its transpose, whose elements are independent normal and chi variables. Various applications of this representation are indicated. Examples are given concerning the diagonal elements of $\mathbf{A}^{-1}$, the sample ordinary and multiple correlation coefficient, the characteristic roots of $\mathbf{A}$ and the sphericity criterion in the bivariate case.