Graphical criteria for positive solutions to linear systems
Abstract
We study linear systems of equations with coefficients in a generic partially ordered ring and a unique solution, and seek conditions for the solution to be nonnegative, that is, every component of the solution is a quotient of two nonnegative elements in . The requirement of a nonnegative solution arises typically in applications, such as in biology and ecology, where quantities of interest are concentrations and abundances. We provide novel conditions on a labeled multidigraph associated with the linear system that guarantee the solution to be nonnegative. Furthermore, we study a generalization of the first class of linear systems, where the coefficient matrix has a specific block form and provide analogous conditions for nonnegativity of the solution, similarly based on a labeled multidigraph. The latter scenario arises naturally in chemical reaction network theory, when studying full or partial parameterizations of the positive part of the steady state variety of a polynomial dynamical system in the concentrations of the molecular species.
Cite
@article{arxiv.1709.01700,
title = {Graphical criteria for positive solutions to linear systems},
author = {Meritxell Sáez and Elisenda Feliu and Carsten Wiuf},
journal= {arXiv preprint arXiv:1709.01700},
year = {2017}
}