The ultradiscrete Toda lattice and the Smith normal form of bidiagonal matrices

Abstract

The discrete Toda lattice preserves the eigenvalues of tridiagonal matrices, and convergence of dependent variables to the eigenvalues can be proved under appropriate conditions. We show that the ultradiscrete Toda lattice preserves invariant factors of a certain bidiagonal matrix over a principal ideal domain and prove convergence of dependent variables to invariant factors using properties of the box and ball system. Using this fact, we present a new method for computing the Smith normal form of a given matrix.