## Condition number of a square matrix with i.i.d. columns drawn from a convex body

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- by Radosław Adamczak, Olivier Guédon, Alexander E. Litvak, Alain Pajor and Nicole Tomczak-Jaegermann PDF
- Proc. Amer. Math. Soc.
**140**(2012), 987-998 Request permission

## Abstract:

We study the smallest singular value of a square random matrix with i.i.d. columns drawn from an isotropic log-concave distribution. An important example is obtained by sampling vectors uniformly distributed in an isotropic convex body. We deduce that the condition number of such matrices is of the order of the size of the matrix and give an estimate on its tail behaviour.## References

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## Additional Information

**Radosław Adamczak**- Affiliation: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
- Email: radamcz@mimuw.edu.pl
**Olivier Guédon**- Affiliation: Université Paris-Est Marne-La-Vallée, Laboratoire d’Analyse et de Mathématiques Appliquées , 5, boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée, Cedex 2, France
- Email: olivier.guedon@univ-mlv.fr
**Alexander E. Litvak**- Affiliation: Department of Mathematics and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1
- MR Author ID: 367520
- Email: alexandr@math.ualberta.ca
**Alain Pajor**- Affiliation: Université Paris-Est Marne-La-Vallée, Laboratoire d’Analyse et de Mathématiques Appliquées, 5, boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée, Cedex 2, France
- Email: Alain.Pajor@univ-mlv.fr
**Nicole Tomczak-Jaegermann**- Affiliation: Department of Mathematics and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1
- MR Author ID: 173265
- Email: nicole.tomczak@ualberta.ca
- Received by editor(s): October 4, 2010
- Received by editor(s) in revised form: December 6, 2010
- Published electronically: June 23, 2011
- Additional Notes: A part of this work was done when the first author held a postdoctoral position at the Department of Mathematical and Statistical Sciences, University of Alberta in Edmonton, Alberta. The position was sponsored by the Pacific Institute for the Mathematical Sciences. Research was partially supported by MNiSW Grant No. N N201 397437 and the Foundation for Polish Science.

The fifth author holds the Canada Research Chair in Geometric Analysis. - Communicated by: Marius Junge
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**140**(2012), 987-998 - MSC (2010): Primary 52A23, 46B06, 60B20, 60E15; Secondary 52A20, 46B09
- DOI: https://doi.org/10.1090/S0002-9939-2011-10994-8
- MathSciNet review: 2869083