On the real Davies’ conjecture

Abstract

We show that every matrix $\mathit{A}\in {\mathbb{R}}^{\mathit{n}\times \mathit{n}}$ is, at least, $\mathit{\delta }\Vert \mathit{A}\Vert$-close to a real matrix $\mathit{A}+\mathit{E}\in {\mathbb{R}}^{\mathit{n}\times \mathit{n}}$ whose eigenvectors have condition number, at most, ${\tilde{\mathit{O}}}_{\mathit{n}}({\mathit{\delta }}^{-1})$. 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.