A Procrustes problem on the Stiefel manifold

Summary.

An orthogonal Procrustes problem on the Stiefel manifold is studied, where a matrix Q with orthonormal columns is to be found that minimizes $\| AQ-B\|_{\rm F}$ for an $l \times m$ matrix A and an $l \times n$ matrix B with $l \geq m$ and $m > n$. Based on the normal and secular equations and the properties of the Stiefel manifold, necessary conditions for a global minimum, as well as necessary and sufficient conditions for a local minimum, are derived.