On eigenvalue gaps of integer matrices

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 (Lu [Minimum eigenvalue separation, ProQuest LLC, Ann Arbor, MI, 1992. Thesis (Ph.D.)–University of California, Berkeley; Wilkinson [The algebraic eigenvalue problem, Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York, 1988]) 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 $\Omega ((2\sqrt {n})^{-n^2}h^{-n^2})$ (Mahler [Michigan Math. J. 11 (1964), pp. 257–262]). 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. Dey et al. [Bit complexity of Jordan normal form and polynomial spectral factorization, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2023, pp. Art. No. 42]).

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 $2^{-n^2/64+o(n^2)}$.