X-simplicity of interval max-min matrices

Ján Plavka; Štefan Berežný

Kybernetika (2018)

  • Volume: 54, Issue: 3, page 413-426
  • ISSN: 0023-5954

Abstract

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A matrix A is said to have 𝐗 -simple image eigenspace if any eigenvector x belonging to the interval 𝐗 = { x : x ̲ ≤ x ≤ x ¯ } containing a constant vector is the unique solution of the system A ⊗ y = x in 𝐗 . The main result of this paper is an extension of 𝐗 -simplicity to interval max-min matrix 𝐀 = { A : A ̲ ≤ A ≤ A ¯ } distinguishing two possibilities, that at least one matrix or all matrices from a given interval have 𝐗 -simple image eigenspace. 𝐗 -simplicity of interval matrices in max-min algebra are studied and equivalent conditions for interval matrices which have 𝐗 -simple image eigenspace are presented. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that have been studied in max-min and max-plus algebras.

How to cite

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Plavka, Ján, and Berežný, Štefan. "X-simplicity of interval max-min matrices." Kybernetika 54.3 (2018): 413-426. <http://eudml.org/doc/294539>.

@article{Plavka2018,
abstract = {A matrix $A$ is said to have $X$-simple image eigenspace if any eigenvector $x$ belonging to the interval $\mbox\{$X$\}=\lbrace x\colon \underline\{x\}\le x\le \overline\{x\}\rbrace $ containing a constant vector is the unique solution of the system $A\otimes y=x$ in $X$. The main result of this paper is an extension of $X$-simplicity to interval max-min matrix $\mbox\{$A$\}=\lbrace A\colon \underline\{A\}\le A\le \overline\{A\}\rbrace $ distinguishing two possibilities, that at least one matrix or all matrices from a given interval have $X$-simple image eigenspace. $X$-simplicity of interval matrices in max-min algebra are studied and equivalent conditions for interval matrices which have $X$-simple image eigenspace are presented. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that have been studied in max-min and max-plus algebras.},
author = {Plavka, Ján, Berežný, Štefan},
journal = {Kybernetika},
keywords = {max-min algebra; interval; eigenspace; simple image set},
language = {eng},
number = {3},
pages = {413-426},
publisher = {Institute of Information Theory and Automation AS CR},
title = {X-simplicity of interval max-min matrices},
url = {http://eudml.org/doc/294539},
volume = {54},
year = {2018},
}

TY - JOUR
AU - Plavka, Ján
AU - Berežný, Štefan
TI - X-simplicity of interval max-min matrices
JO - Kybernetika
PY - 2018
PB - Institute of Information Theory and Automation AS CR
VL - 54
IS - 3
SP - 413
EP - 426
AB - A matrix $A$ is said to have $X$-simple image eigenspace if any eigenvector $x$ belonging to the interval $\mbox{$X$}=\lbrace x\colon \underline{x}\le x\le \overline{x}\rbrace $ containing a constant vector is the unique solution of the system $A\otimes y=x$ in $X$. The main result of this paper is an extension of $X$-simplicity to interval max-min matrix $\mbox{$A$}=\lbrace A\colon \underline{A}\le A\le \overline{A}\rbrace $ distinguishing two possibilities, that at least one matrix or all matrices from a given interval have $X$-simple image eigenspace. $X$-simplicity of interval matrices in max-min algebra are studied and equivalent conditions for interval matrices which have $X$-simple image eigenspace are presented. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that have been studied in max-min and max-plus algebras.
LA - eng
KW - max-min algebra; interval; eigenspace; simple image set
UR - http://eudml.org/doc/294539
ER -

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