Spectral properties of flipped Toeplitz matrices

TL;DR

This paper investigates the spectral properties of flipped Toeplitz matrices, revealing eigenvalue relationships and localization results, with implications for iterative methods solving non-symmetric Toeplitz systems.

Contribution

It establishes eigenvalue sign relationships and localization results for flipped Toeplitz matrices, extending known theorems and aiding convergence analysis of iterative solvers.

Findings

Eigenvalues of flipped Toeplitz matrices relate alternately to those of original matrices.

Localization results for eigenvalues of flipped Toeplitz matrices.

Insights into the convergence of MINRES for non-symmetric Toeplitz systems.

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 $n\times n$ Toeplitz matrix generated by the function $f$ and $Y_n$ is the $n\times n$ 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.