Aaron Abrams (Washington and Lee University): 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?
By an explicit construction using directed graphs, we improve the previously known upper bound on this gap from $h^{-O(n)}$ to $h^{-\frac{n^2}{16} + o(n^2)}$.
Up to a constant in the exponent, this agrees with the known lower bound.
The result is relevant for worst-case analysis of algorithms for diagonalization of integer matrices.

This is joint work with Zeph Landau, Jamie Pommersheim, and Nikhil Srivastava.

Zoom link: https://us06web.zoom.us/j/88002526951?pwd=ZlhlWVpWbTRScEZuVW1USjJJWXgzQT09
Meeting ID: 880 0252 6951
Passcode: 233827