ON EIGENVALUE GAPS OF INTEGER MATRICES

Given an n x 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, 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(-n2/16+o(n2)). Up to a constant in the exponent, this agrees with the known lower bound of Omega((2 root n)(-n2)h(-n2)) (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 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(-n2/32). We also construct 0-1 matrices which have two eigenvalues separated by at most 2(-n2/64+o(n2)).