Eigenspace structure of a max-prod fuzzy matrix

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• 作者：Imran Rashid^a ; ^{imranrashid@comsats.edu.pk" class="auth_mail" title="E-mail the corresponding author} ; Martin Gavalec^b ; ^{martin.gavalec@uhk.cz" class="auth_mail" title="E-mail the corresponding author} ; Richard Cimler^b ; ^{richard.cimler@uhk.cz" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Product triangular norm ; (max ; prod)-algebra ; Fuzzy matrix ; Eigenvector
• 刊名：Fuzzy Sets and Systems
• 出版年：2016
• 出版时间：15 November 2016
• 年：2016
• 卷：303
• 期：Complete
• 页码：136-148
• 全文大小：1182 K

文摘

The eigenvectors of a fuzzy matrix correspond to steady states of a complex discrete-events system, characterized by the given transition matrix and fuzzy state vectors. The descriptions of the eigenspace (the set of all eigenvectors) for matrices in the max-min, max-Łukasiewicz or max-drast fuzzy algebra have been presented in previous papers. The eigenspace of a fuzzy matrix in the max-prod algebra is investigated in this paper. First, necessary and sufficient conditions are shown under which the eigenspace restricted to increasing eigenvectors of a given matrix is non-empty, and the structure of the increasing eigenspace is described. Then, using simultaneous row and column permutations of the matrix, the complete characterization of the whole eigenspace structure of a given fuzzy matrix is shown. The details for matrices of order 3 are only presented. The method works analogously for square matrices of higher orders, with rapidly increasing complexity of the formulas.