Determinant Expansions of Signed Matrices and of Certain Jacobians

TL;DR

This paper investigates the sign patterns in determinants of matrices and Jacobians, providing combinatorial and graph-theoretic methods to analyze their coefficients and sign properties, with applications to chemical network dynamics.

Contribution

It introduces new techniques for counting sign coefficients in determinants and Jacobians, and offers a graph-theoretic test for Jacobian sign patterns in chemical reaction networks.

Findings

Developed a method to count plus and minus coefficients in determinant expansions.

Provided a graph-theoretic criterion for Jacobian sign pattern determination.

Applied results to analyze chemical network Jacobians.

Abstract

This paper treats two topics: matrices with sign patterns and Jacobians of certain mappings. The main topic is counting the number of plus and minus coefficients in the determinant expansion of sign patterns and of these Jacobians. The paper is motivated by an approach to chemical networks initiated by Craciun and Feinberg. We also give a graph-theoretic test for determining when the Jacobian of a chemical reaction dynamics has a sign pattern.