This thesis presents the Lanzos and Arnold method, which represents a technique for calculating several largest or smallest (extreme) eigenvalues of large sparse symmetric matrices, or a set of eigenvalues of non-symmetric matrix. In the first chapter, we introduced notation and displayed the basic properties of the matrix, eigenvectors and their associated eigenvalues. In the second chapter, we elaborate on Lanczo’s method. First, we introduce the notion of Krylov’s subspace and how we compute the orthonormal base. That brings us to a tridiagonal matrix whose eigenvalues approximate the eigenvalues of our initial symmetric matrix. The next step includes a derivation of the Lanczos iteration, after which we describe the errors and convergence of the method. When implemented in computer arithmetic we notice a loss of orthogonality of Lanczos vectors, and therefore we describe Lanczos with complete orthogonalization that corrects the loss of orthogonality. We also mention the selective orthogonalization and the problem of the so-called ghost eigenvalue. The third chapter gives us an insight into Arnold’s algorithm based on Hessenberg’s decomposition, restarted Arnoldi, and asymmetric Lanczos tridiagonalization. In the last chapter, we present a few examples in which we apply the Lanczos algorithm for symmetric matrix.