Linear equations over cones and Collatz–Wielandt numbers Original Research Article

Let K be a proper cone in image, let A be an n×n real matrix that satisfies AKsubset of or equal toK, let b be a given vector of K, and let λ be a given positive real number. The following two linear equations are considered in this paper: (i) (λIn−A)x=b, xset membership, variantK, and (ii) (A−λIn)x=b, xset membership, variantK. 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 λ>ρb(A), and we also find a necessary condition when λ=ρb(A) and also when λ<ρb(A), sufficiently close to ρb(A), where ρb(A) denotes the local spectral radius of A at b. With λ fixed, we also consider the questions of when the set (A−λIn)K∩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.