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All Journal BAREKENG: Jurnal Ilmu Matematika dan Terapan
Siswanto Siswanto
The Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Sebelas Maret, Indonesia
Computer Science & IT Control & Systems Engineering Economics, Econometrics & Finance Energy Engineering Mathematics Mechanical Engineering Physics Transportation

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THE EIGENVECTORS OF REDUCIBLE MATRICES OVER MIN-PLUS ALGEBRA Siswanto Siswanto; Riko Fajarudin
BAREKENG: Jurnal Ilmu Matematika dan Terapan Vol 20 No 4 (2026): BAREKENG: Journal of Mathematics and Its Application
Publisher : PATTIMURA UNIVERSITY

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30598/barekengvol20iss4pp3151-3166

Abstract

The eigenvalues and the eigenvectors of a matrix are fundamental concepts in max-plus algebra, especially for square matrices. A communication graph is classified as strongly connected or not, corresponding to irreducible and reducible matrices, respectively. This study aims to develop methods for determining the eigenvectors of reducible matrices over min-plus algebra. Before determining the eigenvectors, we need to determine the eigenvalues. The methodology used is based on the Frobenius Normal Form and graph condensation to decompose the reducible matrix into spectral classes. The eigenvalues ​​can be found using the graphical method. Eigenvectors can be determined for each corresponding eigenvalue. Once the eigenvectors are known, we can find the set of eigenvectors as a map of a matrix. The main results include a characterization of eigenvector bases for each spectral class, criteria for the existence of finite eigenvectors, and an analysis of computational complexity. An isomorphic correspondence exists between min-plus algebra and max-plus algebra. Hence, the eigenvectors of reducible matrices over min-plus algebra are able to be evaluated according to a concept from eigenvectors of reducible matrices over max-plus algebra. This study is limited to min-plus algebra and does not extend to interval min-plus algebra.
Co-Authors Riko Fajarudin