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Mehtaab Sawhney, MIT, USA
February 9, 2022 @ 6:00 pm - 7:00 pm UTC+4
Title: “On the real Davies’ conjecture”
Abstract: We show that every nxn matrix A is at least h-close to a real nxn matrix A+E whose eigenvectors have condition number at most O(1/h). 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.