Eigenvectors of Hermitian Toeplitz matrices with smooth simple-loop symbols
Eigenvectors of Hermitian Toeplitz matrices with smooth simple-loop symbols
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DOI：
10.1016/j.laa.2015.12.017
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发表时间：
2016-03-15
影响因子：
1.1
通讯作者：
Maximenko, E. A.
中科院分区：
文献类型：
--
作者：
Bogoya, J. M.;Boettcher, A.;Maximenko, E. A.
本文研究了在真实的直线上迹出一个简单环的具有中等光滑符号的Hermitian Toeplitz矩阵的特征向量矩阵的结构和渐近性质。这些结果将已有的带状Toeplitz矩阵的结果推广到第一行和第一列的项具有温和衰减的完全Toeplitz矩阵。我们建立公式的精确和渐近计算的任意规定的个人组成部分的任意规定的个人特征向量。这些公式是在一个函数的维纳-霍普夫因式分解，它只取决于符号和相应的本征值，甚至只取决于这个本征值的近似。本文不旨在提供数值算法。它的主要目的是揭示结构下面的特征向量，这是两个谐波和某些边缘校正的总和。(C)2015 Elsevier Inc. All rights reserved.
The paper is devoted to the structure and the asymptotics of the eigenvector matrix of Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The results extend existing results on banded Toeplitz matrices to full Toeplitz matrices with temperate decay of the entries in the first row and column. We establish formulas for both the exact and the asymptotic computation of arbitrarily prescribed individual components of arbitrarily prescribed individual eigenvectors. These formulas are in terms of the Wiener-Hopf factorization of a function which depends solely on the symbol and the corresponding eigenvalue or even only on an approximation to this eigenvalue. The paper does not aim at providing numerical algorithms. Its main purpose is rather to reveal the structure underneath the eigenvectors, which are shown to be the sum of two harmonics and certain edge corrections. (C) 2015 Elsevier Inc. All rights reserved.