We show that every matrix is, at least, -close to a real matrix whose eigenvectors have condition number, at most, . 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.
V.J. would like to thank Archit Kulkarni and Nikhil Srivastava for introducing him to the problem. The authors would like to thank an anonymous referee for their careful reading of the manuscript and valuable comments.
"On the real Davies’ conjecture." Ann. Probab. 49 (6) 3011 - 3031, November 2021. https://doi.org/10.1214/21-AOP1522