On the real Davies’ conjecture

Abstract

We show that every matrix A∈Rn×n is, at least, δ‖A‖-close to a real matrix A+E∈Rn×n whose eigenvectors have condition number, at most, O˜n(δ−1). In fact, we prove that, with high probability, taking E to be a sufficiently small multiple of an i.i.d. real sub-Gaussian matrix of bounded density suffices. This essentially confirms a speculation of Davies and of Banks, Kulkarni, Mukherjee and Srivastava, who recently proved such a result for i.i.d. complex Gaussian matrices. Along the way we also prove nonasymptotic estimates on the minimum possible distance between any two eigenvalues of a random matrix whose entries have arbitrary means; this part of our paper may be of independent interest.