## Abstract

Let A = (a_{ij}) be an n × n random matrix with i.i.d. entries such that Ea_{11} = 0 and Ea_{11} ^{2} = 1. We prove that for any δ > 0 there is L > 0 depending only on δ, and a subset N of B_{2} ^{n} of cardinality at most exp(δn) such that with probability very close to one we have A(B2n)∪y∈A(N)(y+LnB2n). In fact, a stronger statement holds true. As an application, we show that for some L' > 0 and u ∈ [0, 1) depending only on the distribution law of a11, the smallest singular value sn of the matrix A satisfies P{ s_{n}(A) ≤ εn^{− 1 / 2}} ≤ L' ε+ u^{n} for all ε > 0. The latter result generalizes a theorem of Rudelson and Vershynin which was proved for random matrices with subgaussian entries.

Original language | English (US) |
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Pages (from-to) | 507-544 |

Number of pages | 38 |

Journal | Israel Journal of Mathematics |

Volume | 227 |

Issue number | 2 |

DOIs | |

State | Published - Aug 1 2018 |

## All Science Journal Classification (ASJC) codes

- General Mathematics