THE MATRIX ANGULAR CENTRAL GAUSSIAN DISTRIBUTION

The Riemann space whose elements are m × k (m ≥ k) matrices X such that X′X = Ik is called the Stiefel manifold and denoted by Vk,m. Some distributions on Vk,m, e.g., the matrix Langevin (or von Mises-Fisher) and Bingham distributions and the uniform distribution, have been defined and discussed in the literature. In this paper, we present methods to construct new kinds of distributions on Vk,m and discuss some properties of these distributions. We investigate distributions of the "orientation" HZ = Z(Z′Z) -1 2 (ε{lunate}Vk,m) of an m × k random matrix Z. The general integral form of the density of HZ reduces to a simple mathematical form, when Z has the matrix-variate central normal distribution with parameter Σ, an m × m positive definite matrix. We may call this distribution the matrix angular central Gaussian distribution with parameter Σ, denoted by the MACG (Σ) distribution. The MACG distribution reduces to the angular central Gaussian distribution on the hypersphere for k = 1, which has been already known. Then, we are concerned with distributions of the orientation HY of a linear transformation Y = BZ of Z, where B is an m × m matrix such that ∥B∥ ≠ 0. Utilizing properties of these distributions, we propose a general family of distributions of Z such that HZ has the MACG (Σ) distribution. © 1990.