Spectral properties of flipped Toeplitz matrices

2023/12/11 by Giovanni Barbarino, Barbarino, Giovanni, Sven‐Erik Ekström +9
Computer Science · Mathematics · #15A18 #15B05 #65F10 #Advanced Topics in Algebra #FOS: Mathematics #Holomorphic and Operator Theory #Matrix Theory and Algorithms #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2312.06170

openalex publication_date 2023/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We study the spectral properties of flipped Toeplitz matrices of the form H_n(f)=Y_nT_n(f), where T_n(f) is the n× n Toeplitz matrix generated by the function f and Y_n is the n× n exchange (or flip) matrix having 1 on the main anti-diagonal and 0 elsewhere. In particular, under suitable assumptions on f, we establish an alternating sign relationship between the eigenvalues of H_n(f), the eigenvalues of T_n(f), and the quasi-uniform samples of f. Moreover, after fine-tuning a few known theorems on Toeplitz matrices, we use them to provide localization results for the eigenvalues of H_n(f). Our study is motivated by the convergence analysis of the minimal residual (MINRES) method for the solution of real non-symmetric Toeplitz linear systems of the form T_n(f)\mathbf x=\mathbf b after pre-multiplication of both sides by Y_n, as suggested by Pestana and Wathen.

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