On Eigenvalue Gaps of Integer Matrices

TL;DR

This paper constructs explicit integer matrices with entries in [0,h] that have extremely small eigenvalue gaps, nearly matching known lower bounds, and explores the prevalence of such matrices with small eigenvalue separations.

Contribution

It provides explicit constructions of matrices with minimal eigenvalue gaps close to theoretical lower bounds and shows many matrices also have small eigenvalue gaps.

Findings

Explicit matrices with eigenvalue gaps of at most h^{-n^2/16+o(n^2)}

Many matrices have eigenvalue gaps roughly h^{-n^2/32}

0-1 matrices with eigenvalue gaps at most 2^{-n^2/64+o(n^2)}

Abstract

Given an $n\times n$ matrix with integer entries in the range $[-h,h]$, how close can two of its distinct eigenvalues be? The best previously known examples have a minimum gap of $h^{-O(n)}$. Here we give an explicit construction of matrices with entries in $[0,h]$ with two eigenvalues separated by at most $h^{-n^2/16+o(n^2)}$. Up to a constant in the exponent, this agrees with the known lower bound of $\Omega((2\sqrt{n})^{-n^2}h^{-n^2})$ \cite{mahler1964inequality}. Bounds on the minimum gap are relevant to the worst case analysis of algorithms for diagonalization and computing canonical forms of integer matrices. In addition to our explicit construction, we show there are many matrices with a slightly larger gap of roughly $h^{-n^2/32}$. We also construct 0-1 matrices which have two eigenvalues separated by at most $2^{-n^2/64+o(n^2)}$.