New methods for computing fuzzy eigenvalues and fuzzy eigenvectors of fuzzy matrices using nonlinear programming approach

Abstract

In this paper, we propose a new method to obtain eigenvalues and fuzzy triangular eigenvectors of a fuzzy triangular matrix (Ã), which are the elements of the given fuzzy triangular matrix . To this purpose, we solve the 1 − cut of a fuzzy triangular matrix (Ã) to obtain the 1 − cut of eigenvalues and eigenvectors. Then, based on the results obtained in a 1 − cut mode, we use three new models to determine the left and right widths for those eigenvalues and eigenvectors. So, after some manipulation, in each of the models, the fully fuzzy linear systems (FFLSs) transformed to 2n crisp linear equations and some crisp linear non-equations (that, the first model includes 2(n + 1), the second model includes 2(n + 3) and the third model includes 6n + 2 crisp linear non-equation). Then, we suggest a nonlinear programming problem (NLP) to calculation simultaneous equations and non-equations. Furthermore, we define three other new eigenvalues (namely, fuzzy escribed eigenvalue, fuzzy peripheral eigenvalue, and fuzzy approximate eigenvalue) for a fuzzy triangular matrix (Ã) that does not have any suitable solution. the fuzzy escribed eigenvalue which is placed in a tolerable fuzzy triangular eigenvalue set (TTFES), the fuzzy peripheral eigenvalue placed in a controllable fuzzy triangular eigenvalue set (CTFES), and the fuzzy approximate eigenvalue placed in an approximate fuzzy triangular eigenvalue set (ATFES). Finally, numerical examples are presented to illustrate the proposed method.