User ashin - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:12:57Z http://mathoverflow.net/feeds/user/18036 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83452/convergence-of-eigenvalues Convergence of Eigenvalues Ashin 2011-12-14T18:42:43Z 2012-01-16T20:22:12Z <p>Suppose we have a matrix $A_n = \frac{1}{n}\sum_{i=1}^nX_i X_i^T$, where $X_i$ is a $p$-dimensional random-vector. We also know that $E(XX^T) = \Sigma_{p \times p}$. Let us denote the $j$-th largest eigenvalue of a symmetric positive-definite matrix $M$ by $\lambda_j(M)$. Then can we say anything about convergence of $\lambda_j(A_n) \rightarrow \lambda_j(\Sigma)$ as $n \rightarrow \infty$, that is, whether it converges in probability or in distribution and if so can we characterize the rate of convergence.</p> <p>Thanks a lot, any help is much appreciated.</p> <p>Best Ashin</p> http://mathoverflow.net/questions/77057/partial-derivatives-of-eigen-value-decomposition-or-singular-value-decomposition partial Derivatives of Eigen value decomposition or Singular value decomposition Ashin 2011-10-03T16:51:27Z 2011-10-04T06:11:05Z <p>Hi All,</p> <p>Suppose I've a symmetric matrix $A_{N \times N} = (A_{ij})$ which has a eigen value decomposition $A = UDU'$. I would like to know under what conditions $\frac{\partial U}{\partial A_{ij}}$ exists for all $i,j = 1,2, \ldots, N$. I found the following paper which talks about estimating the Jacobian of the SVD transformation</p> <p><a href="http://www.ics.forth.gr/cvrl/publications/conferences/2000_eccv_SVD_jacobian.pdf" rel="nofollow">http://www.ics.forth.gr/cvrl/publications/conferences/2000_eccv_SVD_jacobian.pdf</a></p> <p>But its not very clear regarding the conditions that the matrix $A$ would need to satisfy. Any help is much appreciated.</p> <p>Thanks Ashin</p> http://mathoverflow.net/questions/76245/measure-of-subspace-of-matrices-with-repeated-singular-values/76260#76260 Answer by Ashin for Measure of Subspace of Matrices with repeated Singular Values Ashin 2011-09-24T03:33:41Z 2011-09-24T03:33:41Z <p>Hi Chris,</p> <p>Thanks a lot for your quick response. But I'm not that familiar with the results that you are using to arrive at the conclusion. Would you kindly be a bit more elaborate. Or maybe you can guide me to the resources where I could find the results that you are using here.</p> <p>Thanks a lot for your help. Ashin</p> http://mathoverflow.net/questions/83452/convergence-of-eigenvalues Comment by Ashin Ashin 2011-12-14T19:59:48Z 2011-12-14T19:59:48Z @ Robert : Yes I do mean $A_n = \frac{1}{n}\sum_{i = 1}^{n}X_i X_i^T$. Sorry about the typo. http://mathoverflow.net/questions/77057/partial-derivatives-of-eigen-value-decomposition-or-singular-value-decomposition/77108#77108 Comment by Ashin Ashin 2011-11-07T03:22:14Z 2011-11-07T03:22:14Z But my question is more about the singular vectors. Are they also going to admit partial derivatives w.r.t. the entries of the matrix almost everywhere. Best