A min-plus analogue of the Jordan canonical form associated with the basis of the generalized eigenspace

Abstract

ABSTRACT In this paper, we investigate a min-plus analogue of Jordan canonical forms of matrices. We first define the generalized eigenvector of a min-plus matrix A as an eigenvector of the kth power of A for some integer k. As in the conventional algebra, we consider a transformation matrix consisting of the basis of the generalized eigenspace. Then, we call a block diagonal matrix obtained from such transformation matrix a Jordan canonical form. In min-plus algebra, however, not all square matrices have Jordan canonical forms. We derive a necessary and sufficient condition of matrices having those in terms of graph theory.