On the real Davies' conjecture

2020/05/18 by Jain, Vishesh, Sah, Ashwin, Sawhney, Mehtaab · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA) #Numerical Analysis (math.NA) #Probability (math.PR) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2005.08908

Abstract

We show that every matrix A ∈ ℝ^{n× n} is at least δ‖A‖-close to a real matrix A+E ∈ ℝ^{n× 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 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.

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