On the real Davies' conjecture

Abstract: We show that every matrix $A \in \mathbb{R}^{n\times n}$ is at least $\delta$|A|$-close to a real matrix$A+E \in \mathbb{R}^{n\times n}$whose eigenvectors have condition number at most$\tilde{O}_{n}(\delta^{-1}).</sup>Infact,weprovethat,withhighprobability,taking.</sup> 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 non-asymptotic 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.