Nonnegativity of the second largest eigenvalue of $4 \times 4$ tridiagonal stochastic matrices
Probability
2026-05-11 v1
Abstract
The spectral study of nonnegative and more specifically stochastic matrices is an important topic in matrix theory. In this paper, we prove a conjecture, formulated by Ran and Teng, which states that the second largest eigenvalue of an irreducible tridiagonal stochastic matrix is nonnegative. We establish this conjecture and extend the result to arbitrary tridiagonal stochastic matrices, including both irreducible and reducible cases.
Cite
@article{arxiv.2605.07591,
title = {Nonnegativity of the second largest eigenvalue of $4 \times 4$ tridiagonal stochastic matrices},
author = {Brando Vagenende and Brecht Verbeken and Marie-Anne Guerry},
journal= {arXiv preprint arXiv:2605.07591},
year = {2026}
}