Maximum Singular Value of a random +1/-1 matrix - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T11:44:36Z http://mathoverflow.net/feeds/question/95632 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95632/maximum-singular-value-of-a-random-1-1-matrix Maximum Singular Value of a random +1/-1 matrix Kostas 2012-05-01T03:11:16Z 2012-05-02T16:20:22Z <p>Hi,</p> <p>Define a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ such that each element is independently and randomly chosen with probability 0.5 to be either +1, or -1. Do you know any result in the literature that talks about properties of this kind of matrices? </p> <p>I have seen that there are some results for other kind of random matrices (for example matrices whose entries are i.i.d gaussian.) but not for this simple matrix of +1/-1.</p> <p>I would be interested for example on the distribution of the $\sigma_{max}(A)$, (not in an asymptotic regime. $m$, $n$ are finite numbers and usually small in my case.)</p> <p>Thank you very much for any pointer or any thoughts.</p> <p>Best,</p> <p>Alex</p> http://mathoverflow.net/questions/95632/maximum-singular-value-of-a-random-1-1-matrix/95666#95666 Answer by Emilio Pisanty for Maximum Singular Value of a random +1/-1 matrix Emilio Pisanty 2012-05-01T12:56:01Z 2012-05-02T16:20:22Z <p>For what it's worth, <a href="http://quicc.net/other_files/svdrandom.cdf" rel="nofollow">here</a>'s some numerical data in Mathematica CDF format.</p> <p>Edit: I've been playing around some more and this <img src="http://quicc.net/other_files/randomsvdhistogram.png" alt="histogram"> is a histogram for 1,000,000 tries with $m=9$, $n=5$ for the five different singular values (each in a different colour). I'm intrigued by the peaks - @Alex, were you expecting that? There is also a significant portion of matrices with one zero singular value, but I am unsure whether it is due to numerical artifacts.</p> http://mathoverflow.net/questions/95632/maximum-singular-value-of-a-random-1-1-matrix/95669#95669 Answer by David Benson-Putnins for Maximum Singular Value of a random +1/-1 matrix David Benson-Putnins 2012-05-01T13:48:28Z 2012-05-01T13:48:28Z <p><a href="http://www-personal.umich.edu/~romanv/papers/non-asymptotic-rmt-plain.pdf" rel="nofollow">http://www-personal.umich.edu/~romanv/papers/non-asymptotic-rmt-plain.pdf</a> Theorem 5.39 (page 23) gives a non-asymptotic upper bound on the largest singular value</p>