Asymptotic diagonalization of matrix systems

JOURNAL OF APPROXIMATION THEORY

卷:171

Article

Kooman, R. J.

关键词: Difference equations;    Perturbations;    Asymptotic behavior;    Levinson-type theorem;    Rescaling;    Poincare-Perron theorem;    Dichotomy condition;   

DOI  :  10.1016/j.jat.2013.03.002

来源: Elsevier

【 摘 要 】

In recent years many results have been obtained on the asymptotic behavior of solutions of the matrix difference equation M(n)x(n) = x(n+1) where {M-n}(n=0)(infinity) is a sequence of k x k-matrices with real or complex entries that are close to diagonal matrices. In this paper we study the question of how to transform a matrix sequence {M-n}(n=0)(infinity) where the entries behave sufficiently regularly, into a sequence of almost-diagonal matrices, so that the results for almost-diagonal matrices can be applied to the difference equation with the transformed sequence. In particular, we Will try to find explicit matrices Bn such that the matrices M-n(1)= Bn+1-1Mn B-n are close to diagonal matrices and a Levinson-type theorem can be applied to transform the sequence {M-n}(n=0)(infinity) into a sequence of diagonal matrices. In the case that the M-n are real 2 x 2-matrices, a fairly general answer is obtained and it is shown how to proceed for a given sequence {M-n}(n=0)(infinity). Furthermore, we prove a couple of results that are useful for the case of general order k. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free