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

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 th...

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Bibliographic Details
Published inLinear & multilinear algebra Vol. 69; no. 15; pp. 2933 - 2943
Main Authors Nishida, Yuki, Sato, Kohei, Watanabe, Sennosuke
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 18.11.2021
Taylor & Francis Ltd
Subjects
Canonical forms
eigenvalue problem
Eigenvectors
Graph theory
Jordan canonical form
Min-plus algebra
Transformations (mathematics)
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Summary: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.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2019.1700892