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Title: On the vectors associated with the roots of max-plus characteristic polynomials (English)
Author: Nishida, Yuki
Author: Watanabe, Sennosuke
Author: Watanabe, Yoshihide
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 65
Issue: 6
Year: 2020
Pages: 785-805
Summary lang: English
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Category: math
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Summary: We discuss the eigenvalue problem in the max-plus algebra. For a max-plus square matrix, the roots of its characteristic polynomial are not its eigenvalues. In this paper, we give the notion of algebraic eigenvectors associated with the roots of characteristic polynomials. Algebraic eigenvectors are the analogues of the usual eigenvectors in the following three senses: (1) An algebraic eigenvector satisfies an equation similar to the equation $A\otimes \boldsymbol {x} = \lambda \otimes \boldsymbol {x}$ for usual eigenvectors. Under a suitable assumption, the equation has a nontrivial solution if and only if $\lambda $ is a root of the characteristic polynomial. (2) The set of algebraic eigenvectors forms a max-plus subspace called algebraic eigenspace. (3) The dimension of each algebraic eigenspace is at most the multiplicity of the corresponding root of the characteristic polynomial. (English)
Keyword: max-plus algebra
Keyword: eigenvalue
Keyword: eigenvector
Keyword: characteristic polynomial
MSC: 15A18
MSC: 15A80
idZBL: Zbl 07285957
idMR: MR4191369
DOI: 10.21136/AM.2020.0374-19
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Date available: 2020-11-18T09:38:57Z
Last updated: 2021-04-08
Stable URL: http://hdl.handle.net/10338.dmlcz/148397
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