Graphical criteria for positive solutions to linear systems

Meritxell Sáez, Elisenda Feliu, Carsten Wiuf

Research output: Indexed journal article Articlepeer-review

6 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)166-193
Number of pages28
JournalLinear Algebra and Its Applications
Publication statusPublished - 1 Sept 2018
Externally publishedYes


  • Chemical reaction networks
  • Linear system
  • Matrix-tree theorem
  • Positive solution
  • Spanning forest
  • Steady state parameterization


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