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In statistics and machine learning, we are interested in the eigenvectors (or singular vectors) of certain matrices (eg covariance matrices, data matrices, etc). However, those matrices are usually perturbed by noises or statistical errors, either from random sampling or structural patterns. The Davis-Kahan theorem is often used to bound the difference between the eigenvectors of a matrix and those of a perturbed matrix , in terms of norm. In this paper, we prove that when is a low-rank and incoherent matrix, the norm perturbation bound of singular vectors (or eigenvectors in the symmetric case) is smaller by a factor of or for left and right vectors, where and are the matrix dimensions. The power of this new perturbation result is shown in robust covariance estimation, particularly when random variables have heavy tails. There, we propose new robust covariance estimators and establish their asymptotic properties using the newly developed perturbation bound. Our theoretical results are verified through extensive numerical experiments.