The Arnoldi Method for Normal Matrices

Abstract

For large Hermitian matrices the preconditionend conjugate gradient algorithm and the Lanczos algorithm are the most important methods for solving linear systems and for computing eigenvalues. There are various generalizations to the nonsymmetric case based on the Arnoldi method or on the nonsymmetric Lanczos algorithm, e.g., generalized minimal residual ((GMRES), residual minimization in a Krylov space) and conjugate gradient squared ((CGS), a biorthogonalization algorithm adapted from the biconjugate gradient method) for linear equations and the incomplete orthogonalization method and the look-ahead Lanczos algorithm for computing eigenvalues. The aim of this paper is to analyse the Arnoldi method applied to a normal matrix. It is shown that for a normal matrix A, the Arnoldi projection $H_p $ is a normal upper Hessenberg matrix if and only if A is (1)-normal. Only in this case, $H_p $ is tridiagonal and the Arnoldi method has the same properties as in the Hermitian case.