Linear Equations over cones and Collatz-Wielandt numbers

Abstract

Let $K$ be a proper cone in $\IR^n$, let $A$ be an $n\times n$ real matrix that satisfies $AK\subseteq K$, let $b$ be a given vector of $K$, and let $\lambda$ be a given positive real number. The following two linear equations are considered in this paper: (i)$(\lambda I_n-A)x=b$, $x\in K$, and (ii)$(A-\lambda I_n)x=b$, $x\in K$. We obtain several equivalent conditions for the solvability of the first equation. For the second equation we give an equivalent condition for its solvability in case when $\lambda>\rho_b (A)$, and we also find a necessary condition when $\lambda=\rho_b (A)$ and also when $\lambda < \rho_b(A)$, sufficiently close to $\rho_b(A)$, where $\rho_b (A)$ denotes the local spectral radius of $A$ at $b$. With $\lambda$ fixed, we also consider the questions of when the set $(A-\lambda I_n)K \bigcap K$ equals $\{0\}$ or $K$, and what the face of $K$ generated by the set is. Then we derive some new results about local spectral radii and Collatz-Wielandt sets (or numbers) associated with a cone-preserving map, and extend a known characterization of $M$-matrices among $Z$-matrices in terms of alternating sequences.