In this paper, we generalize the algorithm described by Rump and Graillat to compute verified and narrow error bounds such that a slightly perturbed matrix is guaranteed to have an eigenvalue with geometric multiplicity $q$ within computed error bounds. The corresponding invariant subspace can be directly obtained by our algorithm. Our verification method is based on border matrix technique. We demonstrate the performance of our algorithm for matrices of dimension up to hundreds with non-defective and defective eigenvalues.