On symmetric hollow integer matrices with eigenvalues bounded from below

TL;DR

This paper investigates the spectral properties of symmetric hollow integer matrices, establishing bounds on eigenvalues and demonstrating the existence of principal submatrices with eigenvalues below a certain threshold for values less than a specific constant.

Contribution

It proves a threshold phenomenon for eigenvalues of symmetric hollow integer matrices, identifying a critical value where the property holds or fails.

Findings

Existence of principal submatrices with eigenvalues below a threshold for  < *

Identification of a critical constant *  2.01980 where the property changes

Bounded eigenvalues for symmetric hollow integer matrices below *

Abstract

A hollow matrix is a square matrix whose diagonal entries are all equal to zero. Define $\lambda^* = \rho^{1/2} + \rho^{-1/2} \approx 2.01980$, where $\rho$ is the unique real root of $x^3 = x + 1$. We show that for every $\lambda < \lambda^*$, there exists $n \in \mathbb{N}$ such that if a symmetric hollow integer matrix has an eigenvalue less than $-\lambda$, then one of its principal submatrices of order at most $n$ does as well. However, the same conclusion does not hold for any $\lambda \ge \lambda^*$.