Eigenvector localizaiton - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:42:14Z http://mathoverflow.net/feeds/question/107689 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107689/eigenvector-localizaiton Eigenvector localizaiton Felix Goldberg 2012-09-20T14:39:03Z 2012-09-20T15:23:09Z <p>I have raised this sort of question <a href="http://mathoverflow.net/questions/98025/concentration-for-eigenvectors" rel="nofollow">before</a> but I think that now I've found a better term for the subject, one which might ring more bells for people - hence the repost. Hope you won't be too angry with me.</p> <p>I am interested in eigenvector localization for deterministic matrices. There is a whole body of work on the random matrix setup but I am interested in bounding the ratio between eigenvector coordinates of certain fixed and messy matrices. Any results out there that you know of pertaining to this? (I know the <a href="http://www.zentralblatt-math.org/zbmath/?index_=2804942&amp;type_=pdf" rel="nofollow">old masters' work</a> on positive/nonnegative matrices but need to handle further cases).</p> http://mathoverflow.net/questions/107689/eigenvector-localizaiton/107694#107694 Answer by Carlo Beenakker for Eigenvector localizaiton Carlo Beenakker 2012-09-20T15:23:09Z 2012-09-20T15:23:09Z <p>some pointers on eigenvector localization, mainly in the context of Perron-Frobenius theory (which in view of your earlier post seems to be what you are looking for):</p> <p><A HREF="http://emis.u-strasbg.fr/journals/ELA/ela-articles/articles/vol16_pp366-379.pdf" rel="nofollow">Principal eigenvectors of irregular graphs</A></p> <p><A HREF="http://www.tandfonline.com/doi/abs/10.1080/03081089508818429" rel="nofollow">Perron-frobenius theory for a generalized eigenproblem</A></p> <p><A HREF="http://epubs.siam.org/doi/abs/10.1137/0124002" rel="nofollow">Bounds for eigenvalues and eigenvectors of a nonnegative matrix which involve a measure of irreducibility</A></p> <p><A HREF="http://epubs.siam.org/doi/abs/10.1137/S0895479893242585" rel="nofollow">A remark on Minc's maximal eigenvector bound for positive matrices</A></p> <p><A HREF="http://www.sciencedirect.com/science/article/pii/S0024379573800060" rel="nofollow">Einschließung des positiven eigenvektors einer nichtnegativen, irreduziblen matrix</A></p>