Matrices that appear in computational mathematics are so often of low rank. Since random ("average") matrices are almost surely of full rank, mathematics needs to explain the abundance of low rank structures. We will give a characterization of certain low rank matrices using Sylvester equations and show that the decay of singular values can be understood via an extremal rational problem. We will use it to explain why low rank matrices appear in galaxy simulations, polynomial interpolation, Krylov methods, and fast transforms.
Alex Townsend is an assistant professor in the Department of Mathematics at Cornell University, with field affiliations in Applied Mathematics and CSE. His research is in spectral element methods, fast transforms, polynomial system solving, and low rank approximation. Prior to coming to Cornell, he was an Applied Math instructor at MIT after completing a D.Phil. at the University of Oxford. He was awarded a Leslie Fox Prize in 2015 for a fast discrete Hankel transform and in 2013 for developing a sparse well-conditioned spectral method for the solution of differential equations.
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