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

TL;DR

This paper proves a conjecture that the second largest eigenvalue of any 4x4 tridiagonal stochastic matrix is nonnegative, extending the result to both irreducible and reducible cases.

Contribution

It establishes and proves a conjecture regarding the nonnegativity of the second largest eigenvalue for all 4x4 tridiagonal stochastic matrices.

Findings

Confirmed the conjecture for irreducible matrices.

Extended the proof to reducible matrices.

Ensured the second largest eigenvalue is nonnegative in all cases.

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.