Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices

TL;DR

This paper investigates the asymptotic behavior of eigenvalues and eigenvectors of Fisher-Hartwig Toeplitz matrices as their size grows large, focusing on a specific parameter regime.

Contribution

It provides a detailed analysis of the eigenstructure of Fisher-Hartwig Toeplitz matrices in the large-size limit, highlighting their asymptotic properties.

Findings

Eigenvalues have a simple asymptotic form as matrix size N approaches infinity.

Eigenvectors exhibit specific structural patterns in the asymptotic limit.

Results are specialized to the case 0<α<|β|<1, simplifying the analysis.

Abstract

A Toeplitz matrix is one in which the matrix elements are constant along diagonals. The Fisher-Hartwig matrices are much-studied singular matrices in the Toeplitz family. The matrices are defined for all orders, $N$. They are parametrized by two constants, $\alpha$ and $\beta$. Their spectrum of eigenvalues has a simple asymptotic form in the limit as $N$ goes to infinity. Here we study the structure of their eigenvalues and eigenvectors in this limiting case. We specialize to the case $0<\alpha<|\beta|<1$, where the behavior is particularly simple.