Spectral properties of flipped Toeplitz matrices and computational applications

Abstract

We study the spectral properties of flipped Toeplitz matrices of the form $H_n\left(f\right)=Y_n{T}_n\left(f\right)$, where $T_n\left(f\right)$ 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 $H_n\left(f\right)$, the eigenvalues of $T_n\left(f\right)$, 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 $H_n\left(f\right)$. 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 $T_n\left(f\right)x=b$ after pre-multiplication of both sides by $Y_n$, as suggested by Pestana and Wathen [26]. A selection of numerical experiments is provided to illustrate the theoretical results and to show how to use the spectral localizations for predicting the MINRES performance on linear systems with coefficient matrix $H_n\left(f\right)$.