Inverse problems for symmetric doubly stochastic matrices whose Sule\u{i}manova spectra are bounded below by 1/2

TL;DR

This paper introduces a new sufficient condition for the spectrum of symmetric doubly stochastic matrices, specifically when eigenvalues are bounded below by 1/2, advancing the spectral inverse problem in this area.

Contribution

It provides a novel criterion for spectra of symmetric doubly stochastic matrices, expanding understanding of the inverse spectral problem with specific eigenvalue bounds.

Findings

Existence of symmetric doubly stochastic matrices with prescribed spectra under new conditions

The criterion is independent of classical conditions like Perfect-Mirsky and Soules

Examples and applications demonstrate the practical relevance of the results

Abstract

A new sufficient condition for a list of real numbers to be the spectrum of a symmetric doubly stochastic matrix is presented; this is a contribution to the classical spectral inverse problem for symmetric doubly stochastic matrices that is still open in its full generality. It is proved that whenever $\lambda_2, \ldots, \lambda_n$ are non-positive real numbers with $1 + \lambda_2 + \ldots + \lambda_n \geqslant 1/2$, then there exists a symmetric, doubly stochastic matrix whose spectrum is precisely $(1, \lambda_2, \ldots, \lambda_n)$. We point out that this criterion is incomparable to the classical sufficient conditions due to Perfect-Mirsky, Soules, and their modern refinements due to Nader et al. We also provide some examples and applications of our results.