On special cases of the generalized max-plus eigenproblem

Abstract

We study the generalized eigenproblem A ⊗ x = λ ⊗ B ⊗ x, where A, B ∈ ℝm×n in the max-plus algebra. It is known that if A and B are symmetric, then there is at most one generalized eigenvalue, but no description of this unique candidate is known in general. We prove that if C = A - B is symmetric, then the common value of all saddle points of C (if any) is the unique candidate for λ. We also explicitly describe the whole spectrum in the case when B is an outer product. It follows that when A is symmetric and B is constant, the smallest column maximum of A is the unique candidate for λ. Finally, we provide a complete description of the spectrum when n = 2.