On Special Cases of the Generalized Max-Plus Eigenproblem

We study the generalized eigenproblem $A\otimes x=\lambda\otimes B\otimes x, $ where $A,B\in\mathbb{R}^{m\times n}$ 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 $\lambda.$ 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 $\lambda.$ Finally, we provide a complete description of the spectrum when $n=2$.