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On symmetric hollow integer matrices with eigenvalues bounded from below

Zilin Jiang

TL;DR

We study symmetric hollow integer matrices and establish a threshold phenomenon for negative eigenvalues. The main result shows that for any $\lambda<\lambda^*$ with $\lambda^* = \rho^{1/2}+\rho^{-1/2}$ and $\rho$ the real root of $x^3=x+1$, there exists a bound $n$ such that any symmetric hollow integer matrix with a smallest eigenvalue less than $-\lambda$ has a principal submatrix of order at most $n$ also with eigenvalue less than $-\lambda$; this bound fails for $\lambda\ge\lambda^*$. The approach connects the problem to signed graphs, classifies minimal forbidden submatrices as $A_{F,a}$ with a signed graph $F$ and a multiplicity function $a$, and applies Dickson's lemma to obtain finiteness for $\lambda<\lambda^*$. The work extends Vijayakumar's result from $\lambda=2$ and relies on finite forbidden-subgraph results for signed graphs, yielding a structural, constructive description of obstructions.

Abstract

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

On symmetric hollow integer matrices with eigenvalues bounded from below

TL;DR

We study symmetric hollow integer matrices and establish a threshold phenomenon for negative eigenvalues. The main result shows that for any with and the real root of , there exists a bound such that any symmetric hollow integer matrix with a smallest eigenvalue less than has a principal submatrix of order at most also with eigenvalue less than ; this bound fails for . The approach connects the problem to signed graphs, classifies minimal forbidden submatrices as with a signed graph and a multiplicity function , and applies Dickson's lemma to obtain finiteness for . The work extends Vijayakumar's result from and relies on finite forbidden-subgraph results for signed graphs, yielding a structural, constructive description of obstructions.

Abstract

A hollow matrix is a square matrix whose diagonal entries are all equal to zero. Define , where is the unique real root of . We show that for every , there exists such that if a symmetric hollow integer matrix has an eigenvalue less than , then one of its principal submatrices of order at most does as well. However, the same conclusion does not hold for any .
Paper Structure (10 sections, 7 theorems, 10 equations)

This paper contains 10 sections, 7 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Proofs
  3. Proof for $\lambda \le 2$
  4. Case 1:
  5. Case 2:
  6. Proof for $\lambda \in (2, \lambda^*)$
  7. Case 1:
  8. Case 2:
  9. Proof for $\lambda \ge \lambda^*$
  10. Concluding remarks

Key Result

Theorem 3

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 $\lambda \ge \lambda^*$.

Theorems & Definitions (16)

  • Definition 1: Minimal forbidden submatrix
  • Definition 2: Minimal forbidden subgraph
  • Theorem 3
  • Lemma 4: Lemma A of Dickson D
  • Definition 5: Switching equivalence
  • Proposition 6
  • Theorem 7: Theorem 1.5 of Jiang and Polyanskii JP
  • Proposition 8
  • proof : Proof of \ref{['thm:main']} for $\lambda \le 2$
  • Remark
  • ...and 6 more