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

TL;DR

This paper introduces a novel ultradiscrete Toda lattice approach that preserves invariant factors of bidiagonal matrices and converges to their Smith normal form, offering a new computational method.

Contribution

It extends the Toda lattice framework to invariant factors, providing a new technique for Smith normal form computation based on ultradiscrete integrable systems.

Findings

Ultradiscrete Toda lattice preserves invariant factors of bidiagonal matrices.

Dependent variables converge to invariant factors using box and ball system properties.

New method for computing Smith normal form of matrices introduced.

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 box and ball system. Using this fact, we present a new method for computing the Smith normal form of a given matrix.