ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

In minisymposium: Tropical Linear Algebra and Beyond

Tue 11:00–11:30, Room AV 01.12
On special cases of the generalized max-plus eigenproblem
Daniel Jones (University of Birmingham)
Joint work with Peter Butkovi\v{c} (University of Birmingham)

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 then the smallest column maximum of $A$ is the unique candidate for $\lambda.$ Finally, we provide a complete description of the spectrum when $n=2$.