  A fundamental class of matrix optimization problems that arise in many areas
of science and engineering is that of quadratic optimization with orthogonality
constraints. Such problems can be solved using line-search methods on the
Stiefel manifold, which are known to converge globally under mild conditions.
To determine the convergence rate of these methods, we give an explicit
estimate of the exponent in a Lojasiewicz inequality for the (non-convex) set
of critical points of the aforementioned class of problems. By combining such
an estimate with known arguments, we are able to establish the linear
convergence of a large class of line-search methods. A key step in our proof is
to establish a local error bound for the set of critical points, which may be
of independent interest.
