Before revision:  This formula is expressed in terms of the perturbation matrix and therefore characterizes how the singular vector perturbation is induced by the additive noise. We then use thisformula to derive a one-sided version of the sin\Theta theorem, as well as improve the bound on the 2,\infty norm of the singular vector perturbation error.
After revision:  These formulae explicitly show that how the perturbation of the original matrix propagates into singular vectors and singular subspaces, thus providing a direct way of analyzing them. Following this, we derive a collection of new results on SVD perturbation related problems, including a tighter bound on the 2,\infty norm of the singular vector perturbation error.