CIS Distinguished Speaker Series

Mark Giesbrecht, Ph.D.

Professor and Director, David R. Cheriton School of Computer Science,
University of Waterloo

April 7, 2017

Time: 10:30 a.m. - Noon
Place: Center for the Arts, Gore Recital Hall

Eigenvalues, elimination and random integer matrices, and some speculative applications to computing with sparse matrices

Abstract: Integer matrices are typically characterized by the "lattice" of combinations of their rows or columns. This is captured nicely by the Smith canonical form, a diagonal matrix of "invariant factors," to which any integer matrix can be transformed through left and right multiplication by unimodular matrices.

But integer matrices can also be viewed as complex matrices, with eigenvalues and eigenvectors, and every such matrix is similar to a unique one in Jordan canonical form.

It would seem a priori that the invariant factors and the eigenvalues would have little to do with each other. Yet we will show that for "almost all" matrices the invariant factors and the eigenvalues are equal under a p-adic valuation, in a very precise sense.

A much-hoped-for link is explored for fast computation of Smith forms of sparse integer matrices, via the better understood algorithms for computing eigenvalues. All the methods are elementary and no particular background beyond linear algebra will be assumed.

Bio: Dr. Mark Giesbrecht is a Professor and Director of the David R. Cheriton School of Computer Science at the University of Waterloo. He received a B.Sc. from the University of British Columbia in 1986, and a Ph.D from the University of Toronto in 1993. He is an ACM Distinguished Scientist, and former Chair of ACM SIGSAM (Special Interest Group on Symbolic and Algebraic Manipulation) and the International Symposium on Symbolic and Algebraic Computation (ISSAC) Steering Committees, as well as serving as ISSAC Program Committee Chair. His research interests are in symbolic computation and computer algebra, as well as computational linear algebra.