Perron-Frobenius theory for a generalized Eigen problem

Abstract

Motivated by economic models, the generalized eigenvalue problem Ax=λ Bx is investigated under the conditions that A is nonnegative and irreducible, there is a nonnegative vector u such that Bu>Au, and b_ij ≤ _ij for all i#j. The last two conditions are equivalent to B-A being a nonsingular M-matrix. The focus is on generalizations of the Perron-Frobenius theory, the classical theory being recovered when B is the identity matrix. These generalizations include identification of a generalized eigenvalue ρ(A,B) in the interval (0,1) with a positive eigenvector, characterizations and easily computable bounds for ρ(A,B), and localization results for all generalized eigenvalues. Dropping the condition that A is irreducible, necessary and sufficient conditions for the problem to have a solution with x≥0 are formulated in terms of basic and final classes, which are natural extensions of these concepts in the classical theory.