The Procrustes Problem for Orthogonal Stiefel Matrices

In this paper we consider the Procrustes problem on the manifold of orthogonal Stiefel matrices. Given matrices ${\cal A}\in {\Bbb R}^{m\times k},$ ${\cal B}\in {\Bbb R}^{m\times p},$ $m\ge p \ge k,$ we seek the minimum of $\|{\cal A}-{\cal B}Q\|^2$ for all matrices $Q\in {\Bbb R}^{p\times k},$ $Q^TQ=I_{k\times k}$. We introduce a class of relaxation methods for generating sequences of approximations to a minimizer and offer a geometric interpretation of these methods. Results of numerical experiments illustrating the convergence of the methods are given.