Comments on: Outliers in the spectrum of iid matrices with bounded rank perturbations http://terrytao.wordpress.com/2010/12/22/outliers-in-the-spectrum-of-iid-matrices-with-bounded-rank-permutations/ Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Wed, 19 Jun 2013 09:15:53 +0000 hourly 1 http://wordpress.com/ By: Matrix identities as derivatives of determinant identities « What’s new http://terrytao.wordpress.com/2010/12/22/outliers-in-the-spectrum-of-iid-matrices-with-bounded-rank-permutations/#comment-213090 Sun, 13 Jan 2013 20:36:46 +0000 http://terrytao.wordpress.com/?p=4542#comment-213090 [...] for some . (This type of situation is also common in random matrix theory, for instance it arose in this previous paper of mine on outliers to the circular law.) If is invertible, then from (1) and (2) one [...]

]]> By: Char http://terrytao.wordpress.com/2010/12/22/outliers-in-the-spectrum-of-iid-matrices-with-bounded-rank-permutations/#comment-168108 Sun, 09 Sep 2012 01:17:58 +0000 http://terrytao.wordpress.com/?p=4542#comment-168108 Hi Terry, I have a question about the distribution of the outliers. According to Silverstein’s paper, outliers are N~(mu*n,sigma^2) given that the outliers are not normalised. I ran a simulation with R. I tried different values of n, mu, sigma and roughly 1000 matrices. I got the mean of the outliers of mu*n but I got the standard deviation approximately sigma*sqrt(2) instead of sigma. Is it a correct result?

]]> By: Yashar http://terrytao.wordpress.com/2010/12/22/outliers-in-the-spectrum-of-iid-matrices-with-bounded-rank-permutations/#comment-51893 Wed, 20 Apr 2011 00:27:08 +0000 http://terrytao.wordpress.com/?p=4542#comment-51893 I guess I found my own mistake: the Laurent series has infinite zeros actually…

]]> By: Yashar http://terrytao.wordpress.com/2010/12/22/outliers-in-the-spectrum-of-iid-matrices-with-bounded-rank-permutations/#comment-51892 Wed, 20 Apr 2011 00:24:34 +0000 http://terrytao.wordpress.com/?p=4542#comment-51892 Hi Terry,

I have a question about the non-self-adjoint case that was studied by Rajan and Abbott. Simulations suggest that the total number of outlier eigenvalues (at least in the “balanced” case where the components of \psi_n sum up to 0) scales like \sqrt(n) (up to n ~ 3000). But you say above that the outlier point process converges to the zeros of that random Laurent series which is independent of n, suggesting that the number of outliers is O(1), and not O(\sqrt(n)), and my simulations are not actually showing the very large n behavior.
Am I right (in concluding the number is O(1) based on what you say), or am I missing something?
I would really appreciate your answer.
Thanks.

]]> By: andrescaicedo http://terrytao.wordpress.com/2010/12/22/outliers-in-the-spectrum-of-iid-matrices-with-bounded-rank-permutations/#comment-49644 Mon, 17 Jan 2011 16:17:08 +0000 http://terrytao.wordpress.com/?p=4542#comment-49644 Thanks!

]]> By: Terence Tao http://terrytao.wordpress.com/2010/12/22/outliers-in-the-spectrum-of-iid-matrices-with-bounded-rank-permutations/#comment-49641 Mon, 17 Jan 2011 13:00:41 +0000 http://terrytao.wordpress.com/?p=4542#comment-49641 I discussed this identity at

http://terrytao.wordpress.com/2010/12/17/the-mesoscopic-structure-of-gue-eigenvalues/

]]> By: andrescaicedo http://terrytao.wordpress.com/2010/12/22/outliers-in-the-spectrum-of-iid-matrices-with-bounded-rank-permutations/#comment-49637 Mon, 17 Jan 2011 06:47:13 +0000 http://terrytao.wordpress.com/?p=4542#comment-49637 Terry, didn’t you have a post discussing identity (1)? Seems to have disappeared. (Maybe I’m confused?)

]]> By: Terence Tao http://terrytao.wordpress.com/2010/12/22/outliers-in-the-spectrum-of-iid-matrices-with-bounded-rank-permutations/#comment-49380 Wed, 05 Jan 2011 05:46:01 +0000 http://terrytao.wordpress.com/?p=4542#comment-49380 Thanks for the reference!

]]> By: Raj Rao http://terrytao.wordpress.com/2010/12/22/outliers-in-the-spectrum-of-iid-matrices-with-bounded-rank-permutations/#comment-49352 Mon, 03 Jan 2011 17:30:12 +0000 http://terrytao.wordpress.com/?p=4542#comment-49352 (Or at least its analogue in the symmetric case)

]]> By: Raj Rao http://terrytao.wordpress.com/2010/12/22/outliers-in-the-spectrum-of-iid-matrices-with-bounded-rank-permutations/#comment-49351 Mon, 03 Jan 2011 17:28:46 +0000 http://terrytao.wordpress.com/?p=4542#comment-49351 In the paper:
Restricted rank modification of the symmetric eigenvalue problem: Theoretical considerations by Arbenz, Gander, Golub

the formula in your Remark 2.2 is referred to as the “modified Weinstein determinant” following Weinstein and Stenger’s “Methods for Intermediate Problems for Eigenvalues”.

Just FYI :-)

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