A Procrustes Problem on the Stiefel Manifold

Lars Eld\'en and Haesun Park

ABSTRACT

An orthogonal Procrustes problem on the Stiefel manifold is studied,
where a matrix $Q$ with orthonormal columns is to be found that
minimizes $\| AQ-B\|_F$ 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.