Cayley-Hamilton theorem in the min-plus algebra

Abstract

Max-plus algebra is the set of ℝ ϵ = ℝ ∪ {ϵ} where ℝ is a set of all real numbers and ϵ = -∞ which is equipped with max (⊕) and plus (⊗) operations. The structure of max-plus algebra is semi-field. Max-plus algebra has structural similarities to conventional algebra. Because of these similarities, concept in conventional algebra such as the Cayley-Hamilton theorem has max-algebraic equivalence. Another semi-field is min-plus algebra, so that the Cayley-Hamilton theorem also has a min-plus algebraic equivalence. In this paper, we will show how the Cayley-Hamilton theorem can be proved in the min-plus algebra.