Tolerance problems for generalized eigenvectors of interval fuzzy matrices

Gavalec, Martin, et al. "Tolerance problems for generalized eigenvectors of interval fuzzy matrices." Kybernetika 58.5 (2022): 760-778. <http://eudml.org/doc/299470>.

@article{Gavalec2022,
abstract = {Fuzzy algebra is a special type of algebraic structure in which classical addition and multiplication are replaced by maximum and minimum (denoted $ \oplus $ and $ \otimes $, respectively). The eigenproblem is the search for a vector $x$ (an eigenvector) and a constant $\lambda $ (an eigenvalue) such that $A\otimes x=\lambda \otimes x$, where $A$ is a given matrix. This paper investigates a generalization of the eigenproblem in fuzzy algebra. We solve the equation $A\otimes x = \lambda \otimes B\otimes x$ with given matrices $A,B$ and unknown constant $\lambda $ and vector $x$. Generalized eigenvectors have interesting and useful properties in the various computational tasks with inexact (interval) matrix and vector inputs. This paper studies the properties of generalized interval eigenvectors of interval matrices. Three types of generalized interval eigenvectors: strongly tolerable generalized eigenvectors, tolerable generalized eigenvectors and weakly tolerable generalized eigenvectors are proposed and polynomial procedures for testing the obtained equivalent conditions are presented.},
author = {Gavalec, Martin, Myšková, Helena, Plavka, Ján, Ponce, Daniela},
journal = {Kybernetika},
keywords = {interval generalized eigenvector; fuzzy matrix},
language = {eng},
number = {5},
pages = {760-778},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Tolerance problems for generalized eigenvectors of interval fuzzy matrices},
url = {http://eudml.org/doc/299470},
volume = {58},
year = {2022},
}

TY - JOUR
AU - Gavalec, Martin
AU - Myšková, Helena
AU - Plavka, Ján
AU - Ponce, Daniela
TI - Tolerance problems for generalized eigenvectors of interval fuzzy matrices
JO - Kybernetika
PY - 2022
PB - Institute of Information Theory and Automation AS CR
VL - 58
IS - 5
SP - 760
EP - 778
AB - Fuzzy algebra is a special type of algebraic structure in which classical addition and multiplication are replaced by maximum and minimum (denoted $ \oplus $ and $ \otimes $, respectively). The eigenproblem is the search for a vector $x$ (an eigenvector) and a constant $\lambda $ (an eigenvalue) such that $A\otimes x=\lambda \otimes x$, where $A$ is a given matrix. This paper investigates a generalization of the eigenproblem in fuzzy algebra. We solve the equation $A\otimes x = \lambda \otimes B\otimes x$ with given matrices $A,B$ and unknown constant $\lambda $ and vector $x$. Generalized eigenvectors have interesting and useful properties in the various computational tasks with inexact (interval) matrix and vector inputs. This paper studies the properties of generalized interval eigenvectors of interval matrices. Three types of generalized interval eigenvectors: strongly tolerable generalized eigenvectors, tolerable generalized eigenvectors and weakly tolerable generalized eigenvectors are proposed and polynomial procedures for testing the obtained equivalent conditions are presented.
LA - eng
KW - interval generalized eigenvector; fuzzy matrix
UR - http://eudml.org/doc/299470
ER -