A Note on Eigenvalue of Matrices over The Symmetrized Max-Plus Algebra

Abstract

Max-plus algebra is the structure that doesn’t have an inverse of additive. Therefore, there exists an equation that doesn’t have a solution. For example, equation 3 ®x=2 has no solution because there is no x such that max(3,x) = 2. The max-plus will have an inverse element of addition if that structure is extended to the symmetrized max-plus algebra. The expansion into a larger system is the same as the expansion of the natural number into an integer number. This paper describes the necessary or sufficient condition of the eigenvalue of matrices over the symmetrized max-plus algebra using the linear balance systems A®xV b with V as a balance relation.