Description:
<p> Let<tex-math>$V_{k,m}$</tex-math>denote the Stiefel manifold which consists of m × k(m ≥ k) matrices X such that<tex-math>$X^{\prime }X=I_{k}$</tex-math>. We present decompositions of a random matrix X and then of the invariant measure on<tex-math>$V_{k,m}$</tex-math>, relative to a fixed subspace ν in R<sup>m</sup>, for all possible four cases to be considered according to the sizes of k, m, and the dimension of v. The results are utilized for deriving the distributions of the canonical correlation coefficients between two random matrices of "general" dimensions, and for discussing high dimensional limit theorems (as m → ∞) on<tex-math>$V_{k,m}$</tex-math>. </p>