Random matrix with non-identical variances - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T20:45:27Z http://mathoverflow.net/feeds/question/89425 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89425/random-matrix-with-non-identical-variances Random matrix with non-identical variances unknown (google) 2012-02-24T17:51:23Z 2012-05-20T20:35:17Z <p>Hello,</p> <p>Consider $A$ a $n \times n$ random matrix with centered Gaussian entries $A_{i,j}$ such that $$\mathbb{E}[A_{i,j}^2]=\sigma_j^2/n$$. The variances depend on the column only.</p> <p>What do we know on the eigenvalues distribution ? In particular, if one assumes that $$\frac{1}{n}\sum_{j=1}^n \sigma_j^2 \to \bar{\sigma}^2$$ then is it true that :</p> <p>$\rho(A):=max(|\lambda_k|) \to \bar{\sigma}^2$ when $n\to \infty$ ?</p> http://mathoverflow.net/questions/89425/random-matrix-with-non-identical-variances/97499#97499 Answer by Daniel for Random matrix with non-identical variances Daniel 2012-05-20T19:16:19Z 2012-05-20T19:16:19Z <p>I'm not sure this answers your question exactly but there are some known bounds. For example, <a href="http://www.ams.org/journals/proc/2005-133-05/S0002-9939-04-07800-1/home.html" rel="nofollow">Latala</a> provides the following <em>high probability bound</em> under the conditions of finite fourth moments for the matrix entries $$\|A\|_2 \leq C \left( \max_i \sqrt{\sum_j \mathbb{E}A_{ij}^2}+ \max_j \sqrt{\sum_i \mathbb{E}A_{ij}^2} + \sqrt[4]{\sum_{i,j}\mathbb{E}A_{ij}^4} \right).$$</p> <p>It would also be worthwhile to check out some of the recent review papers. One is by Laszlo Erdos and another is by Terrence Tao. </p> http://mathoverflow.net/questions/89425/random-matrix-with-non-identical-variances/97504#97504 Answer by Terry Tao for Random matrix with non-identical variances Terry Tao 2012-05-20T20:35:17Z 2012-05-20T20:35:17Z <p>Your matrix model can be written as the product of a standard gaussian matrix with a deterministic diagonal matrix. As such, I believe that Theorem 4 of this paper of Bordenave: <a href="http://arxiv.org/abs/1010.3087" rel="nofollow">http://arxiv.org/abs/1010.3087</a> should give the asymptotic empirical spectral distribution under some reasonable hypotheses on the variances.</p>