An ergodic theorem for classes of preconditioned matrices

Abstract

We consider abstract classes of matrices { A } satisfying some structural conditions and, in particular, satisfying a crucial assumption about the asymptotic distribution of eigenvalues. We prove a similar distribution property for classes of preconditioned matrices constructed by using representants of { A }. As a particular case, this result applies to preconditioned matrices coming from several important contexts: Finite Differences and Faedo-Ritz-Galerkin linear systems associated with elliptic and semielliptic boundary value problems, very general Hermitian Toeplitz structures generated by multivariate L l functions. This result answers in the positive some structural questions raised by Tyrtyshnikov [E. Tyrtyshnikov, Linear Algebra Appl. 207 (1994) 225–249] and by the author [S. Serra, Linear Algebra Appl. 267 (1997) 139–161; S. Serra, SIAM J. Numer. Anal., in press] in the Toeplitz context.