The characteristic equation of the matrix over min-plus algebra

Abstract

Max-plus algebra is one of many idempotent semi-rings. The max-plus algebraic structure is semi field while the conventional algebra is a field. Because of their similar structure, various properties and concepts in the conventional algebra such as characteristic equations have max plus algebraic equivalence. The characteristic equation has been proved in the max-plus algebra. The other semi-field is min-plus algebra. Because of the structure in the min-plus algebra is also similar to the conventional algebra, the characteristic equation also has a min-plus algebraic equivalent. In this paper, it is discussed how to prove the characteristic equation of the matrix over conventional algebra into the min-plus algebra. The results are almost the same. The addition and multiplication operations in the conventional algebra are replaced by min and plus operations in the min-plus algebra. In addition, because of the min-plus algebra does not define the subtraction operation, the formulation of the characteristic equation of the matrix over min-plus algebra is not equal to zero.