Computing the eigenvalues and eigenvectors of a fuzzy matrix

Abstract

Computation of fuzzy eigenvalues and fuzzy eigenvectors of a fuzzy matrix is a challenging problem. Determining the maximal and minimal symmetric solution can help to find the eigenvalues. So, we try to compute these eigenvalues by determining the maximal and minimal symmetric solution of the fully fuzzy linear system A X = λ X.

Krylov subspace method for fuzzy eigenvalue problem

“…Therefore from (8) and (9), it can be respectively shown that Based on similar arguments given in [4,49], set…”

“…However, the explicit form of the quantities α l, z ts and α u, z ts with respect to the different mentioned nature of the fuzzy matrix A(r) = { a ij (r)} will clearly be different. For further information, interested readers may consider the following papers [4,5,8,49]. Indeed in §5, numerical examples are illustrated for the different cases of the coefficient matrix.…”

“…Fuzzy eigenvectors of a real matrix and the structure of fuzzy eigenspaces had been studied in [57], together with a relationship with the real eigenspaces. In [49], via a 2-step procedure, the authors computed the maximal and minimal symmetric solution of the FEVP. Lately in [3], Allahviranloo and Hooshangian proposed a bilateral scheme, where the 1-cut of the problem is first solved and then unknown symmetric spreads are allocated to each row of the 1-cut system.…”

Fuzzy Eigenvalues and Fuzzy Eigenvectors of Fuzzy Markov Chain Transition Matrix under Max-min Composition

“…The few existing studies on this topic provide partial solutions in a very restrictive framework, where the fuzzy matrices considered are subject to several constraints (see Buckley [1]; Salahshour and al. [2]). In addition to this, it should be stressed that those papers only treat fuzzy matrices with fuzzy coefficients, and they use the fuzzy arithmetic.…”

“…[5]; Salahshour and al. [2] studied the fuzzy eigenvalue and the fuzzy eigenvector by using the maximal and minimal symmetric spreads. Based on the definition of fuzzy Markov chain introduced by Avrachenkov and Sanchez [6], we address the same problem, but in a context where the fuzzy matrix has the same form as the one used in [6].…”