  This paper is on the normal approximation of singular subspaces when the
noise matrix has i.i.d. entries. Our contributions are three-fold. First, we
derive an explicit representation formula of the empirical spectral projectors.
The formula is neat and holds for deterministic matrix perturbations. Second,
we calculate the expected projection distance between the empirical singular
subspaces and true singular subspaces. Our method allows obtaining arbitrary
$k$-th order approximation of the expected projection distance. Third, we prove
the non-asymptotical normal approximation of the projection distance with
different levels of bias corrections. By the $\lceil \log(d_1+d_2)\rceil$-th
order bias corrections, the asymptotical normality holds under optimal
signal-to-noise ration (SNR) condition where $d_1$ and $d_2$ denote the matrix
sizes. In addition, it shows that higher order approximations are unnecessary
when $|d_1-d_2|=O((d_1+d_2)^{1/2})$. Finally, we provide comprehensive
simulation results to merit our theoretic discoveries.
  Unlike the existing results, our approach is non-asymptotical and the
convergence rates are established. Our method allows the rank $r$ to diverge as
fast as $o((d_1+d_2)^{1/3})$. Moreover, our method requires no eigen-gap
condition (except the SNR) and no constraints between $d_1$ and $d_2$.
