Invertibility via distance for random matrices with continuous distributions

Let A be an n by n random matrix with independent centered rows, so that each row has real-valued components, is isotropic and log-concave. Further, let M be any fixed n by n real matrix. We derive small ball probability estimates for the smallest singular value of the non-centered random matrix A+M. Our method is free from any use of covering arguments, and is principally different from a standard (by now) approach involving a decomposition of the unit sphere and coverings, as well as from an approach of Sankar-Spielman-Teng for non-centered Gaussian matrices. Our method gives an estimate for the condition number of A+M which essentially matches the known bounds for a non-centered Gaussian matrix.

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