Inspired by the practical importance of social networks, economic networks, biological networks and so on, studies on large and complex networks have attracted a surge of attention in the recent years. Link prediction is a fundamental issue to understand the mechanisms by which new links are added to the networks. We introduce the method of robust principal component analysis (robust PCA) into link prediction, and estimate the missing entries of the adjacency matrix. On the one hand, our algorithm is based on the sparse and low-rank property of the matrix, while, on the other hand, it also performs very well when the network is dense. This is because a relatively dense real network is also sparse in comparison to the complete graph. According to extensive experiments on real networks from disparate fields, when the target network is connected and sufficiently dense, whether it is weighted or unweighted, our method is demonstrated to be very effective and with prediction accuracy being considerably improved compared to many state-of-the-art algorithms.