Examples Using Differential Forms

Abstract

In this chapter we calculate probability density functions for the canonical correlations (Section 9.1), Hotelling T² (Section 9.2), and the eigenvalues of the sample covariance matrix (Section 9.3). The calculations of Section 9.1 were stated by James (1954) who did the problem in the central case only. The noncentral case was computed by Constantine (1963). Our derivation differs somewhat from that of Constantine in that we place more emphasis on the use of differential forms. The results of Section 9.3 are taken directly from James (1954). The calculations of Section 9.2 are original to the author and are inserted in order to include this important example. In the problems, Section 9.4, several problems present background material. Problems 9.4.7 and 9.4.8 treat the distribution of the correlation coefficients using differential forms. There are other examples of the use of differential forms in the book, notably Section 5.3 on the eigenvalues of the covariance matrix, which should be compared with Section 9.3, Section 10.3 on the noncentral multivariate beta density function, which should be compared with Section 5.7, and Section 11.0.1 on the decomposition X = AT^t, T ∈ T(k) and A a n × k matrix such that A^tA = I _k .