Decay Rates for Inverses of Band Matrices

Abstract

Abstract. Spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices. The rate of decay oí A'1 can be bounded in terms of the (essential) spectrum of A A* for general A and in terms of the (essential) spectrum of A for positive definite A. In the positive definite case the bound can be attained. These results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices. We also establish decay rates for certain generalized inverses.1. Introduction. The exponential decay of the entries of inverse of band matrices has been of some use in establishing local rates of convergence of spline approximations [12], [11], [6] and in bounding the L^-norm of the orthogonal projection onto spline spaces [4] and [15]. Kershaw proved a result of this nature for tridiagonal matrices and Descloux's paper [7] contains such a result for Grammian matrices arising in finite element approximations although exponential decay is not explicitly mentioned. For general banded invertible matrices the first proof appeared in [6]. A later proof in [3], [5] gave explicit estimates for the rate of decay. In this paper we use spectral theory and a result of Chebyshev on the best approximation of (x -a)~l by polynomials to give a new proof. The bounds on the rate of decay obtained from this proof appear to be sharper than those previously known and are actually attained in some cases. In addition, the method of proof easily extends to certain generalized inverses and certain nonbanded matrices. We show that the rate of decay for A ~l given by our method depends on only the essential spectrum of AA* and is, thus, stable under banded compact perturbations. This fact is used to establish the exponential decay of the eigenvectors of certain generalized eigenvalue

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