What's in a random integer matrix?
Mark Giesbrecht (Cheriton School of Computer Science, University of Waterloo)
Thursday, 8 May 2014, 15:00, b-it 1.25 (cosec meeting room)
Integer matrices are typically characterized by the lattice of linear combinations of their rows or columns. This is captured in the diagonal Smith canonical form, a diagonal matrix of "invariant factors", to which any integer matrix can be transformed by left and right multiplication by unimodular matrices. But integer matrices can also be viewed as complex matrices, with eigenvalues, eigenvectors and a Jordan canonical form unique up to similarity. A priori, the invariant factors and the eigenvalues would seem to have nothing 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, at all sufficiently large primes p. Experimental results will be shown for smaller primes. All the methods are elementary and no particular background beyond linear algebra will be assumed.
This is joint work with Mustafa Elsheikh.