Uniform distribution on a Stiefel manifold

The uniform distribution on a Stiefel manifold is a matrix-variate distribution that plays an important role in multivariate statistics. There one often encounters integrals over the orthogonal group or over the Stiefel manifold with respect to an invariant measure. For example, this distribution arises in the study of the functional determinant under transformations involving orthogonal or semi-orthogonal matrices. The uniform distribution on the Stiefel manifold corresponds to the normalized Haar measure on the Stiefel manifold.

A random matrix uniformly distributed on the Stiefel manifold is invariant under the two-sided group action of the product ${\displaystyle O(p)\times O(n)}$ of orthogonal groups, i.e. ${\displaystyle X\sim V_{1}XV_{2}}$ for all ${\displaystyle V_{1}\in O(p)}$ and ${\displaystyle V_{2}\in O(n)}$.