New methods are introduced for deriving the sampling distributions of statistics obtained from a normal multivariate population. Exterior differential forms are used to represent the invariant measures on the orthogonal group and the Grassmann and Stiefel manifolds. The first part is devoted to a mathematical exposition of these. In the second part, the theory is applied; first, to the derivation of the distribution of the canonical correlation coefficients when the corresponding population parameters are zero; and secondly, to split the distribution of a normal multivariate sample into three independent distributions, (a) essentially the Wishart distribution, (b) the invariant distribution of a random plane which is given by the invariant measure on the Grassmann manifold, (c) the invariant distribution of a random orthogonal matrix. This decomposition provides derivations of the Wishart distribution and of the distribution of the latent roots of the sample variance covariance matrix when the population roots are equal.
A. T. James. "Normal Multivariate Analysis and the Orthogonal Group." Ann. Math. Statist. 25 (1) 40 - 75, March, 1954. https://doi.org/10.1214/aoms/1177728846