Characterizing matrices with 𝐗 -simple image eigenspace in max-min semiring

Ján Plavka; Sergeĭ Sergeev

Kybernetika (2016)

  • Volume: 52, Issue: 4, page 497-513
  • 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 ¯ } is the unique solution of the system A ⊗ y = x in 𝐗 . The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. 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 Sergeev, Sergeĭ. "Characterizing matrices with ${\bf {X}}$-simple image eigenspace in max-min semiring." Kybernetika 52.4 (2016): 497-513. <http://eudml.org/doc/286782>.

@article{Plavka2016,
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 $ is the unique solution of the system $A\otimes y=x$ in $\mbox\{$X$\}$. The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. 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, Sergeev, Sergeĭ},
journal = {Kybernetika},
keywords = {max-min algebra; interval; weakly robust; weakly stable; eigenspace; simple image set; max-min algebra; interval; weakly robust; weakly stable; eigenspace; simple image set},
language = {eng},
number = {4},
pages = {497-513},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Characterizing matrices with $\{\bf \{X\}\}$-simple image eigenspace in max-min semiring},
url = {http://eudml.org/doc/286782},
volume = {52},
year = {2016},
}

TY - JOUR
AU - Plavka, Ján
AU - Sergeev, Sergeĭ
TI - Characterizing matrices with ${\bf {X}}$-simple image eigenspace in max-min semiring
JO - Kybernetika
PY - 2016
PB - Institute of Information Theory and Automation AS CR
VL - 52
IS - 4
SP - 497
EP - 513
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 $ is the unique solution of the system $A\otimes y=x$ in $\mbox{$X$}$. The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. 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; weakly robust; weakly stable; eigenspace; simple image set; max-min algebra; interval; weakly robust; weakly stable; eigenspace; simple image set
UR - http://eudml.org/doc/286782
ER -

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