On the real Davies' conjecture
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We show that every matrix A ∈R^n× n is at least δ A-close to a real matrixA+E ∈R^n× nwhose eigenvectors have condition number at mostÕ_n(δ^-1). In fact, we prove that, with high probability, takingEto 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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