  This paper studies the Schatten-$q$ error of low-rank matrix estimation by
singular value decomposition under perturbation. We specifically establish a
perturbation bound on the low-rank matrix estimation via a perturbation
projection error bound. Then, we establish lower bounds to justify the
tightness of the upper bound on the low-rank matrix estimation error. We
further develop a user-friendly sin$\Theta$ bound for singular subspace
perturbation based on the matrix perturbation projection error bound. Finally,
we demonstrate the advantage of our results over the ones in the literature by
simulation.
