Decay rates for inverses of band matrices 论文

摘要

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 of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Superscript negative 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{A^{ - 1}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be bounded in terms of the (essential) spectrum of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper A Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>A</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">A{A^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for general <italic>A</italic> and in terms of the (essential) spectrum of <italic>A</italic> for positive definite <italic>A</italic> . 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.