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Invertibility of random matrices: norm of the inverse. (English) Zbl 1175.15030
Let $$A$$ be an $$n \times n$$ matrix, whose entries are independent copies of a centered random variable satisfying the subgaussian tail estimate. The author proves that the operator norm of $$A^{-1}$$ does not exceed $$C n^{3/2}$$ with probability close to 1. For random matrices a polynomial bound was unknown. Proving such a polynomial estimate is the main aim of this paper. The paper contains the sections of Introduction, Overview of the proof, Preliminary results, Halasz type lemma, Small ball probability estimates, Singular profile, and Proof of Theorem 1.1.

##### MSC:
 15B52 Random matrices (algebraic aspects) 46B09 Probabilistic methods in Banach space theory 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 15A09 Theory of matrix inversion and generalized inverses
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