This paper establishes the superconvergence and the related recovery type a posteriori error estimators based on projection method for finite element approximation of the elliptic eigenvalue problems. The projection method is a postprocessing procedure that constructs a new approximation by using the least squares method. The results are based on some regularity assumption for the elliptic problem, and are applicable to the finite element approximations of self-adjoint elliptic eigenvalue problems with general quasi-regular partitions. Therefore, the result of this paper can be employed to provide useful a posteriori error estimators in adaptive finite element computation under unstructured meshes. Key words: finite element approximation, projection method, eigenvalue problems, a posteriori error estimate, superconvergence