Estimating the distances of the shortest path between given pairs of nodes in a graph is a basic operation in a wide variety of applications including social network analysis, web retrieval, etc. Such applications require a response on the order of a few milliseconds, but exact algorithms to compute the distance of the shortest path exactly do not work on real-world large-scale networks, because of their infeasible time complexities. The landmark-based methods approximate distances by using a few nodes as landmarks, and can accurately estimate shortest-path distances with feasible time complexities. However, they fail at estimating small distances, as it is difficult for a few selected landmarks to cover the shortest paths of many close node pairs. To tackle this problem, we present a novel method EigenSP, that estimates the shortest-path distance by using an adjacency matrix approximated by a few eigenvalues and eigenvectors. The average relative error rate of EigenSP is lower than that of the landmark-based methods on large graphs with many short distances. Empirical results suggest that EigenSP estimates small distances better than the landmark-based methods.