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 the eigenvalues and fuzzy triangular eigenvectors of a fuzzy triangular matrix $\left( {\tilde{A}} \right)$, where the elements of the fuzzy triangular matrix are given. For this purpose, we solve 1-cut of a fuzzy triangular matrix $\left( {\tilde{A}} \right)$ to obtain 1-cut of eigenvalues and eigenvectors. Considering the interval system $\left[ {\tilde{A}} \right]_{\alpha } \left[ {\tilde{X}} \right]_{\alpha } = \left[ {\tilde{\lambda }} \right]_{\alpha } \left[ {\tilde{X}} \right]_{\alpha } 0 \le \alpha \le 1$ as α-cut of the fuzzy system $\tilde{A}\tilde{X} = \tilde{\lambda }\tilde{X}$, to determine the left and right width of eigenvalues $\left[ {\tilde{\lambda }} \right]_{\alpha }$ and eigenvector elements $\left[ {\tilde{X}} \right]_{\alpha } 0 \le \alpha \le 1$, we make a system of linear and nonlinear equations and inequalities. And we propose nonlinear programming models to solve the system of linear and nonlinear equations and inequalities and to calculate $\left[ {\tilde{\lambda }} \right]_{\alpha }$ and $\left[ {\tilde{X}} \right]_{\alpha } 0 \le \alpha \le 1$. 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 the fuzzy eigenvalue and fuzzy eigenvector cannot be obtained based on interval calculations. Therefore, the fuzzy escribed eigenvalue which is placed in a tolerable fuzzy triangular eigenvalue set, the fuzzy peripheral eigenvalue placed in a controllable fuzzy triangular eigenvalue set, and the fuzzy approximate eigenvalue placed in an approximate fuzzy triangular eigenvalue set is defined in this paper. Finally, numerical examples are presented to illustrate the proposed method.