On the vectors associated with the roots of max-plus characteristic polynomials

Nishida, Yuki, Watanabe, Sennosuke, and Watanabe, Yoshihide. "On the vectors associated with the roots of max-plus characteristic polynomials." Applications of Mathematics 65.6 (2020): 785-805. <http://eudml.org/doc/297270>.

@article{Nishida2020,
abstract = {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 $A\otimes x = \lambda \otimes x$ 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.},
author = {Nishida, Yuki, Watanabe, Sennosuke, Watanabe, Yoshihide},
journal = {Applications of Mathematics},
keywords = {max-plus algebra; eigenvalue; eigenvector; characteristic polynomial},
language = {eng},
number = {6},
pages = {785-805},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the vectors associated with the roots of max-plus characteristic polynomials},
url = {http://eudml.org/doc/297270},
volume = {65},
year = {2020},
}

TY - JOUR
AU - Nishida, Yuki
AU - Watanabe, Sennosuke
AU - Watanabe, Yoshihide
TI - On the vectors associated with the roots of max-plus characteristic polynomials
JO - Applications of Mathematics
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 6
SP - 785
EP - 805
AB - 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 $A\otimes x = \lambda \otimes x$ 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.
LA - eng
KW - max-plus algebra; eigenvalue; eigenvector; characteristic polynomial
UR - http://eudml.org/doc/297270
ER -