  We study a continuous-time system that solves optimization problems over the
set of orthonormal matrices, which is also known as the Stiefel manifold. The
resulting optimization flow follows a path that is not always on the manifold
but asymptotically lands on the manifold. We introduce a generalized Stiefel
manifold to which we extend the canonical metric of the Stiefel manifold. We
show that the vector field of the proposed flow can be interpreted as the sum
of a Riemannian gradient on a generalized Stiefel manifold and a normal vector.
Moreover, we prove that the proposed flow globally converges to the set of
critical points, and any local minimum and isolated critical point is
asymptotically stable.
