Smallest singular value of a random rectangular matrix | Zendy

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Smallest singular value of a random rectangular matrix

Author(s) -

Rudelson Mark,

Vershynin Roman

Publication year - 2009

Publication title -

communications on pure and applied mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 3.12

H-Index - 115

eISSN - 1097-0312

pISSN - 0010-3640

DOI - 10.1002/cpa.20294

Subject(s) - mathematics , singular value , independent and identically distributed random variables , random matrix , gaussian , matrix (chemical analysis) , order (exchange) , value (mathematics) , circular law , combinatorics , random variable , statistics , eigenvalues and eigenvectors , sum of normally distributed random variables , physics , materials science , finance , quantum mechanics , economics , composite material

We prove an optimal estimate of the smallest singular value of a random sub‐Gaussian matrix, valid for all dimensions. For an N × n matrix A with independent and identically distributed sub‐Gaussian entries, the smallest singular value of A is at least of the order √ N − √ n − 1 with high probability. A sharp estimate on the probability is also obtained. © 2009 Wiley Periodicals, Inc.

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