Controllable and tolerable generalized eigenvectors of interval max-plus matrices

Matej Gazda; Ján Plavka

Kybernetika (2021)

  • Volume: 57, Issue: 6, page 922-938
  • ISSN: 0023-5954

Abstract

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By max-plus algebra we mean the set of reals ℝ equipped with the operations a ⊕ b = max { a , b } and a ⊗ b = a + b for a , b ∈ ℝ . A vector x is said to be a generalized eigenvector of max-plus matrices A , B ∈ ℝ ( m , n ) if A ⊗ x = λ ⊗ B ⊗ x for some λ ∈ ℝ . The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval) entries are studied and complete solutions of the controllable, the tolerable and the strong generalized eigenproblem in max-plus algebra are presented. As a consequence of the obtained results, efficient algorithms for checking equivalent conditions are introduced.

How to cite

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Gazda, Matej, and Plavka, Ján. "Controllable and tolerable generalized eigenvectors of interval max-plus matrices." Kybernetika 57.6 (2021): 922-938. <http://eudml.org/doc/298014>.

@article{Gazda2021,
abstract = {By max-plus algebra we mean the set of reals $\mathbb \{R\}$ equipped with the operations $a\oplus b=\max \lbrace a,b\rbrace $ and $a\otimes b= a+b $ for $a,b\in \mathbb \{R\}.$ A vector $x$ is said to be a generalized eigenvector of max-plus matrices $A, B\in \mathbb \{R\}(m,n)$ if $A\otimes x=\lambda \otimes B\otimes x$ for some $\lambda \in \mathbb \{R\}$. The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval) entries are studied and complete solutions of the controllable, the tolerable and the strong generalized eigenproblem in max-plus algebra are presented. As a consequence of the obtained results, efficient algorithms for checking equivalent conditions are introduced.},
author = {Gazda, Matej, Plavka, Ján},
journal = {Kybernetika},
keywords = {interval generalized eigenvector; fuzzy matrix},
language = {eng},
number = {6},
pages = {922-938},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Controllable and tolerable generalized eigenvectors of interval max-plus matrices},
url = {http://eudml.org/doc/298014},
volume = {57},
year = {2021},
}

TY - JOUR
AU - Gazda, Matej
AU - Plavka, Ján
TI - Controllable and tolerable generalized eigenvectors of interval max-plus matrices
JO - Kybernetika
PY - 2021
PB - Institute of Information Theory and Automation AS CR
VL - 57
IS - 6
SP - 922
EP - 938
AB - By max-plus algebra we mean the set of reals $\mathbb {R}$ equipped with the operations $a\oplus b=\max \lbrace a,b\rbrace $ and $a\otimes b= a+b $ for $a,b\in \mathbb {R}.$ A vector $x$ is said to be a generalized eigenvector of max-plus matrices $A, B\in \mathbb {R}(m,n)$ if $A\otimes x=\lambda \otimes B\otimes x$ for some $\lambda \in \mathbb {R}$. The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval) entries are studied and complete solutions of the controllable, the tolerable and the strong generalized eigenproblem in max-plus algebra are presented. As a consequence of the obtained results, efficient algorithms for checking equivalent conditions are introduced.
LA - eng
KW - interval generalized eigenvector; fuzzy matrix
UR - http://eudml.org/doc/298014
ER -

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