Nonnegativity of the second largest eigenvalue of $4 \times 4$ tridiagonal stochastic matrices

Brando Vagenende; Brecht Verbeken; Marie-Anne Guerry

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arXiv · 2605.07591

Nonnegativity of the second largest eigenvalue of $4 \times 4$ tridiagonal stochastic matrices

Brando Vagenende·Brecht Verbeken·Marie-Anne Guerry

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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 $4\times4$ tridiagonal stochastic matrix is nonnegative. We establish this conjecture and extend the result to arbitrary $4\times4$ tridiagonal stochastic matrices, including both irreducible and reducible cases.

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Brando Vagenende, Brecht Verbeken, Marie-Anne Guerry. 2026-05-08. Nonnegativity of the second largest eigenvalue of $4 \times 4$ tridiagonal stochastic matrices. https://arxiv.org/abs/2605.07591

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