The structured distance to singularity of a symmetric tridiagonal Toeplitz matrix

TL;DR

This paper derives explicit formulas for the structured distance to singularity of symmetric tridiagonal Toeplitz matrices, analyzing their spectral sensitivity to structured perturbations to identify the closest singular matrix.

Contribution

It provides the first explicit formulas for the structured distance to singularity of symmetric tridiagonal Toeplitz matrices, enhancing understanding of their spectral stability.

Findings

Explicit formulas for structured distance to singularity

Analysis of spectral sensitivity to structured perturbations

Identification of the closest singular matrix within the structured class

Abstract

This paper is concerned with the distance of a symmetric tridiagonal Toeplitz matrix $T$ to the variety of similarly structured singular matrices, and with determining the closest matrix to $T$ in this variety. Explicit formulas are presented, that exploit the analysis of the sensitivity of the spectrum of $T$ with respect to structure-preserving perturbations of its entries.