Perron-frobenius theory for a generalized eigenproblem

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Abstract
Motivated by economic models, the generalized eigenvalue problem Ax=λBx is investigated under the conditions that A is nonnegative and irreducible, there is a nonnegative vector u such that Bu>Au, and bij ⩽ 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.

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