Eigenvalues of partial differential equations and their numerical approximations are important research subjects, simply because of their wide application in sciences. We propose to enhance the eigenvalue approximation by post-processing the numerical data from a general purpose finite element software. This prost-processing procedure for the second-order differential operators is called gradient recovery and its computational complexity is of order O(N), where N is the number of unknowns. Both self-adjoint and non-self-adjoint operators will be discussed.