Before revision:  This expression clearly characterizes the linear dependence of the sin\Theta angle on the perturbation matrix. It serves as a tool that allows us to study a set of related problems in a more systematical way.In particular, we derive a useful perturbation bound for the Principal Component Analysis , which seems to be missing in the literature.
After revision:  This expression clearly characterizes the linear dependence of the sin\Theta theorem, as well as improve the bound on the \ell_{2,\infty Principal Component Analysis , which seems to be missing in the literature.