Before revision: In this paper , we establish an exact expression of the sin\Theta angle for the singular subspace perturbation. This expression clearly characterizes the linear dependence of the sin\Theta angle on the perturbation matrix.
After revision: In this paper , we establish an exact expression of the sin\Theta angles between the perturbed and the original singular subspaces. 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 this formula to derive a one-sided version of the sin\Theta angle on the perturbation matrix.