TY - JOUR

T1 - Graphical criteria for positive solutions to linear systems

AU - Sáez, Meritxell

AU - Feliu, Elisenda

AU - Wiuf, Carsten

N1 - Funding Information:
This work was funded by the Danish Research Council and the Lundbeck Foundation , Denmark.
Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2018/9/1

Y1 - 2018/9/1

N2 - We study linear systems of equations with coefficients in a generic partially ordered ring R 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 R. The requirement of a nonnegative solution arises typically in applications, for example 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.

AB - We study linear systems of equations with coefficients in a generic partially ordered ring R 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 R. The requirement of a nonnegative solution arises typically in applications, for example 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.

KW - Chemical reaction networks

KW - Linear system

KW - Matrix-tree theorem

KW - Positive solution

KW - Spanning forest

KW - Steady state parameterization

UR - http://www.scopus.com/inward/record.url?scp=85046166876&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2018.04.014

DO - 10.1016/j.laa.2018.04.014

M3 - Article

AN - SCOPUS:85046166876

SN - 0024-3795

VL - 552

SP - 166

EP - 193

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -