Abstract: Integer matrices are typically characterized by the lattice of linear 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.All the methods are elementary and no particular background beyond linear algebra will be assumed.  A much-hoped-for link and some open problems on algorithms for sparse integer matrices will be explored.This is joint work with graduate student Mustafa Elsheikh

 

Bio: Dr. Mark Giesbrecht is Director and Professor of the 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, computer algebra, finite fields, and computational linear algebra.  He serves on the editorial board of the Journal of Symbolic Computation.