Characteristic polynomial of matrices over interval max-plus algebra

Abstract

Let ℝ be the set of all real number and e =-∞. Max-plus algebra is the set ℝe = ℝ ∪ {e} that is equipped with maximum (⊕) and addition (⊗) operations. For each A∈ℝen×n can be found its eigenvalue and its eigenvector. The problem how to find eigenvalue and eigenvector is eigenproblem. A number of methods have been developed to determine the eigenvalues of a matrix over max-plus algebra. One of them is by using characteristic polynomial. Max-plus algebra has been expanded into interval max-plus algebra. Interval max-plus algebra is the set I(ℝ)e={x=[x_, x¯]| x_, x¯∈ℝ,e<x_≤x¯)∪{e} and e = [e,e] which is equipped with maximum (⊕¯) and addition (⊗¯) operations. Let A∈I(ℝ)en×n, the eigenproblem in interval max-plus algebra is A⊗¯x=λ⊗¯x where λ∈I(ℝ) and x∈I(ℝ)en respectively are eigenvalue and eigenvector for matrices A. In this research, we developed a method to find the characteristic polynomial of matrices over interval max-plus algebra and its eigenvalue.