On Eigenvalue Gaps of Integer Matrices

12/14/2022

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by   Aaron Abrams, et al.

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Given an n× 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 Ω((2√(n))^-n^2h^-n^2) <cit.>. Bounds on the minimum gap are relevant to the worst case analysis of algorithms for diagonalization and computing canonical forms of integer matrices (e.g. <cit.>). 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).