Determining the Inverse of a Matrix over Min-Plus Algebra

Abstract

Linear algebra over the semiring  R_ε with ⊗ (plus) and ⨁ (maximum) operations which is known as max-plus algebra. One of the isomorphic with this algebra is a min-plus algebra. Min-plus algebra that is the set R_(ε^' )=R∪{ε'}, with ⊗^' (plus) and ⨁' (minimum) operations. Given a matrix whose components are elements of R_(ε^' )  is called min-plus algebra matrices. Any matrix can be connected by an inverse. In conventional algebra, a square matrix is said an invertible matrix if the det⁡〖(A)〗≠0. In contrast to max-plus algebra, a matrix is said to have inverse condition if it meets certain conditions. Some concepts from the max-plus algebra can be transformed to the min-plus, because of their structural similarity. This means that the inverse matrix concept in max-plus can be constructed into a min-plus version. Thus, this study will explain the inverse of a matrix over the min-plus algebra, property of multiplying two invertible matrices, and connection between invertible matrix and linear mapping used the literature study method, with literature sources such as books, journals, articles, and theses. The data analysis technique used in this research is qualitative data analysis technique. Then, this article has a principal result that is matrix A∈R_(ε^')^(n×n) has a right inverse if and only if there are permutations of σ and the value of λ_i<ε', i∈{1,2,3,…,n} such that A=P_σ ⊗^' D(λ_i ) which is the inverse of matrices. Furthermore, if B is the correct inverse that satisfies A⊗^' B=E then B⊗^' A=E and B is uniquely determined by A.