On Systems of Linear Equations with Nonnegative Coefficients

Abstract

We consider a system of linear equations with positive coefficients, where the entries of the nonnegative irreducible coefficient matrix depend on a parameter vector. We say that the parameter vector is feasible if there exists a positive solution to this system. A set of all feasible parameter vectors is called the feasibility set. If all the positive entries are log-convex functions, the paper shows that the associated Perron root is log-convex on the parameter set and the l1-norm of the solution is log-convex on the feasibility set. These results imply that the feasibility set is a convex set regardless whether the l1-norm of the solution is bounded by some positive real number or not. Finally, we show important applications of these results to wireless communication networks and prove some other interesting results for this special case.