We study robust PCA for the fully observed setting, which is about separating a low rank matrix L and a sparse matrix S from their sum D=L+S. In this talk, a new algorithm, dubbed accelerated alternating projections, is introduced for robust PCA which significantly improves the computational efficiency of the existing non-convex algorithms. Exact recovery guarantee has been established which shows linear convergence of the proposed algorithm. Empirical performance evaluations confirm the advantage of our algorithm over other state-of-the-art algorithms for robust PCA. Furthermore, we extend our method to the low-rank Hankel matrix, with its application to the spectrally sparse signals.