Similarity and isospectral transformation of max-plus matrices (Recent developments in mathematics of integrable systems)

Abstract

Max-plus algebra is a semiring with addition “max” and multiplication “+”. It has been used to describe ultradiscrete integrable systems. In this paper, we discuss transformations of max-plus matrices. We first define the similarity transformation and show that such transformation preserves the maximum eigenvalue of the matrix. Unlike the case of conventional linear algebra, the similarity transformation does not induce an equivalence relation on maxplus matrices. We develop the concept of unitary-pair semigroups so that the transformation becomes symmetric and transitive.