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Mathematics Colloquium

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Date:  Tuesday, October 11, 2011

Title:  Random matrices: invertibility, structure, and applications

Abstract:  At the heart of random matrix theory lies the realization that the spectrum of a random matrix H stabilizes as the dimensions of H grow to infinity. This phenomenon is captured by the limit laws of random matrix theory, in particular by Wigner's semicircle law, Girko's circular law, and Marchenko-Pastur law. These limit laws offer us a clear global and asymptotic picture of the spectrum of H. In the last few years, a considerable progress was made on the more difficult local and non-asymptotic regimes. In the non-asymptotic regime, the dimensions of H are fixed rather than grow to infinity. In the local regime, one zooms in on a small part of the spectrum of H until one sees individual eigenvalues. We will see some examples of heuristics, problems and results of the non-asymptotic random matrix theory. We will also discuss lots of examples and applications in various fields -- dimension reduction in computer science and signal processing, linear solvers in numerical analysis, spectra of nuclei in physics, and especially the principal component analysis in statistics.


Speaker:  Roman Vershynin
Institution:  U of Michigan

 

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