Smallest singular value of sparse random matrices

Volume 212 / 2012

Alexander E. Litvak, Omar Rivasplata Studia Mathematica 212 (2012), 195-218 MSC: 46B06, 60B20, 15B52. DOI: 10.4064/sm212-3-1

Abstract

We extend probability estimates on the smallest singular value of random matrices with independent entries to a class of sparse random matrices. We show that one can relax a previously used condition of uniform boundedness of the variances from below. This allows us to consider matrices with null entries or, more generally, with entries having small variances. Our results do not assume identical distribution of the entries of a random matrix and help to clarify the role of the variances of the entries. We also show that it is enough to require boundedness from above of the $r$th moment, $r > 2$, of the corresponding entries.

Authors

  • Alexander E. LitvakDepartment of Mathematics and Statistical Sciences
    University of Alberta
    Edmonton, Alberta T6G 2G1, Canada
    e-mail
  • Omar RivasplataDepartment of Mathematics and Statistical Sciences
    University of Alberta
    Edmonton, Alberta T6G 2G1, Canada
    e-mail

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