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.