Geodesic
On the vectors associated with the roots of max-plus characteristic polynomials
Applications of Mathematics, Tome 65 (2020) no. 6, pp. 785-805 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.21136/AM.2020.0374-19
Classification : 15A18, 15A80
Keywords: max-plus algebra; eigenvalue; eigenvector; characteristic polynomial 
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Nishida, Yuki; Watanabe, Sennosuke; Watanabe, Yoshihide. On the vectors associated with the roots of max-plus characteristic polynomials. Applications of Mathematics, Tome 65 (2020) no. 6, pp. 785-805. doi: 10.21136/AM.2020.0374-19

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