Distributions of orientations on Stiefel manifolds

The Riemann space whose elements are m - k (m [greater, double equals] k) matrices X, i.e., orientations, such that X'X = Ik is called the Stiefel manifold Vk,m. The matrix Langevin (or von Mises-Fisher) and matrix Bingham distributions have been suggested as distributions on Vk,m. In this paper, we present some distributional results on Vk,m. Two kinds of decomposition are given of the differential form for the invariant measure on Vk,m, and they are utilized to derive distributions on the component Stiefel manifolds and subspaces of Vk,m for the above-mentioned two distributions. The singular value decomposition of the sum of a random sample from the matrix Langevin distribution gives the maximum likelihood estimators of the population orientations and modal orientation. We derive sampling distributions of matrix statistics including these sample estimators. Furthermore, representations in terms of the Hankel transform and multi-sample distribution theory are briefly discussed.