In this paper, we consider the discrete Legendre projection methods to solve the eigenvalue problem. Using sufficiently accurate numerical quadrature rule, we obtain the error bounds for gap between the spectral subspaces, eigenvalues and iterated eigenvectors for the eigenvalue problem in norm. We also obtain the superconvergence results for eigenvalues and iterated eigenvectors in discrete Legendre Galerkin methods. Numerical examples are presented to illustrate the theoretical results.