  In this paper, we establish a useful set of formulae for the $\sin\Theta$
distance between the original and the perturbed singular subspaces. 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
$\ell_{2,\infty}$ norm of the singular vector perturbation errors under
Gaussian noise, a new stability analysis of the Principal Component Analysis
and an error bound on the singular value thresholding operator. For the latter
two, we consider the most general rectangular matrices with full matrix rank.
