TAILIEUCHUNG - Đề tài " Invertibility of random matrices: norm of the inverse "

Let A be an n × n matrix, whose entries are independent copies of a centered random variable satisfying the subgaussian tail estimate. We prove that the operator norm of A−1 does not exceed Cn3/2 with probability close to 1. 1. Introduction Let A be an n × n matrix, whose entries are independent, identically distributed random variables. The spectral properties of such matrices, in particular invertibility, have been extensively studied (see, . [M] and the survey [DS]). While A is almost surely invertible whenever its entries are absolutely continuous, the case of discrete entries is highly nontrivial. . | Annals of Mathematics Invertibility of random matrices norm of the inverse By Mark Rudelson Annals of Mathematics 168 2008 575 600 Invertibility of random matrices norm of the inverse By Mark Rudelson Abstract Let A be an n X n matrix whose entries are independent copies of a centered random variable satisfying the subgaussian tail estimate. We prove that the operator norm of A-1 does not exceed Cn3 2 with probability close to 1. 1. Introduction Let A be an n X n matrix whose entries are independent identically distributed random variables. The spectral properties of such matrices in particular invertibility have been extensively studied see . M and the survey DS . While A is almost surely invertible whenever its entries are absolutely continuous the case of discrete entries is highly nontrivial. Even in the case when the entries of A are independent random variables taking values 1 with probability 1 2 the precise order of probability that A is singular is unknown. Komlos K1 K2 proved that this probability is o 1 as n 1. This result was improved by Kahn Komlós and Szemeredi KKS who showed that this probability is bounded above by 0n for some absolute constant 0 1. The value of 0 has been recently improved in a series of papers by Tao and Vu TV1 TV2 to 0 3 4 o 1 the conjectured value is 0 1 2 o 1 . However these papers do not address the quantitative characterization of invertibility namely the norm of the inverse matrix considered as an operator from R to R . Random matrices are one of the standard tools in geometric functional analysis. They are used in particular to estimate the Banach-Mazur distance between finite-dimensional Banach spaces and to construct sections of convex bodies possessing certain properties. In all these questions condition number or the distortion IIAk A-1 I plays the crucial role. Since the norm of A is usually highly concentrated the distortion is determined by the norm of A-1. The estimate of the norm of A-1 is known only in the case

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