Available online at www.ispacs.com/jfsva Volume 2012, Year 2012 Article ID jfsva-00120, 18 pages doi:10.5899/2012/jfsva-00120 Research Article Computing the eigenvalues and eigenvectors of a fuzzy matrix S. Salahshour 1,∗ R. Rodr´ıguez-L´opez 2, F. Karimi 3, A. Kumar 4 1 Young Researchers Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran 2 Departamento de An´alisisMatem´atico, Facultad de Matem´aticas, Universidad de Santiago de Compostela, 15782, Santiago de Compostela, Spain 3 Department of Mathematics, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran 4 School of Mathematics and Computer Applications, Thapar University, Patiala, India Copyright 2012 ⃝c S. Salahshour, R. Rodr´ıguez-L´opez, F. Karimi and A. Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited. 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 AeXe = λeXe. Keywords : Fully Fuzzy Linear System, Maximal symmetric solution, Minimal symmetric solution. 1 Introduction The problem of finding eigenvalues arises in a wide variety of practical applications. It arises in almost all branches of science and engineering. Many important characteristics of physical and engineering systems, such as stability, can often be determined just by knowing the nature and location of the eigenvalues. The aim of this paper is to determine fuzzy eigenvalues and fuzzy eigenvectors of a fuzzy matrix Ae. e e e The system of linear equations AX = ~b, where the elements,a ~ij, of the matrix A and the elements, ~bi, of the vector ~b are fuzzy numbers, is called a fully fuzzy linear system (FFLS). Although FFLSs have been studied by many authors, a solution method for finding fuzzy eigenvalues and fuzzy eigenvectors of a fuzzy matrix Ae with fuzzy idempotent has ∗Corresponding author. Email address: [email protected] Tel:+989359130694 1 not been given yet. Recently, Allahviranloo et al. [1] proposed a novel method to solve a FFLS based on the 1-cut expansion. In that method, some spreads and, therefore, some new solutions are derived which are placed in TSS or CSS. We use this method to find fuzzy eigenvalues and fuzzy eigenvectors of a fuzzy matrix Ae by transforming the system AeXe = λeX:e (1.1) The outline of the paper is as follows: In first place, the system AeXe = λeXe is solved in a 1-cut position, obtaining a crisp system. Next, some unknown symmetric spreads are allocated to each row of the 1-cut system. Afterwards, we analyze the structure of the obtained spreads, which are not neces- sarily linear. We study the eigenvalues and eigenvectors obtained and see what symmetric spreads allow to obtain minimal or maximal pairs of eigenvalues and eigenvectors. In section 2, the most important notations used are mentioned. Then, in Section 3, we present our new method to obtain eigenvalues and eigenvectors. Besides, we discuss the reasons why the spreads obtained provide maximal- and minimal symmetric solutions. Also, section 4 gives a numerical example. The final section ends this paper with a brief conclusion. 2 Preliminaries First, we recall some definitions concerning fuzzy numbers: Definition 2.1. Fuzzy numbers are one way to describe the vagueness and lack of precision of data. The theory of fuzzy numbers is based on the theory of fuzzy sets which Zadeh [2] introduced in 1965. The concept of a fuzzy number was first used by Nahmias in the United States and by Dubois and Prade in France in the late 1970s [3]. Definition 2.2.[1] A fuzzy number u in parametric form is a pair (u; u) of functions u(r), u(r), 0 ≤ r ≤ 1, which satisfy the following requirements: 1. u(r) is a bounded non-decreasing left continuous function in (0; 1], and right contin- uous at 0, 2. u(r) is a bounded non-increasing left continuous function in (0; 1], and right contin- uous at 0, 3. u(r) ≤ u(r), 0 ≤ r ≤ 1. The triangular fuzzy number u = (x0; σ; β) (fuzzy number with a defuzzifier x0, left fuzziness σ > 0 and right fuzziness β > 0) is a fuzzy set where the membership function is as follows 8 > 1 (x − x + σ); x − σ ≤ x ≤ x ; > σ 0 0 0 <> 1 (x − x + β); x ≤ x ≤ x + β; u(x) = > β 0 0 0 > :> 0; otherwise. It can easily be verified that r [u] = [x0 − (1 − r)σ; x0 + (1 − r)β] = [x0 − α1(r); x0 + α2(r)]; 8r 2 (0; 1]; 2 where α1(r) = (1 − r)σ and α2(r) = (1 − r)β. Given a fuzzy number u, the support of u is defined in the following way: supp(u) = fx j u(x) > 0g, where fx j u(x) > 0g represents the closure of the set fx j u(x) > 0g. Given two arbitrary fuzzy numbersu ~(r) = [u(r); u(r)] andv ~(r) = [v(r); v(r)], we define addition, subtraction and multiplication as follows: 1. Addition: (~u +v ~)(r) = [u(r) + v(r); u(r) + v(r)], for 0 ≤ r ≤ 1. 2. Subtraction: (~u − v~)(r) = [u(r) − v(r); u(r) − v(r)], for 0 ≤ r ≤ 1. 3. Multiplication: (~uv~)(r) = [minfu(r)v(r); u(r)v(r); u(r)v(r); u(r)v(r)g; maxfu(r)v(r); u(r)v(r); u(r)v(r); u(r)v(r)g]; for 0 ≤ r ≤ 1. e Definition 2.3. Let A = (~aij) be a fuzzy matrix. A fuzzy number λ and a fuzzy vector e t ≤ ≤ ≤ ≤ X = (~x1;:::; x~n) , given byx ~i(r) = [xi(r); xi(r)], 1 i n, 0 r 1, are called, respectively, the eigenvalue and eigenvector of matrix Ae if Xn Xn Xn Xn a~ijx~j = a~ijx~j = λx~i; a~ijx~j = a~ijx~j = λx~i: j=1 j=1 j=1 j=1 Definition 2.4.[4] The united solution set (USS), the tolerable solution set (TSS) and the controllable solution set (CSS) for a fully fuzzy system AX = b are, respectively, the following sets: 0 n 0 0 0 0 0 0 n 0 X99 = fx 2 R :(9A 2 A)(9b 2 b) s:t: A x = b g = fx 2 R : Ax \ b =6 ;g; 0 n 0 0 0 0 0 0 n 0 X89 = fx 2 R :(8A 2 A)(9b 2 b) s:t: A x = b g = fx 2 R : Ax ⊆ bg; 0 n 0 0 0 0 0 0 n 0 X98 = fx 2 R :(8b 2 b)(9A 2 A) s:t: A x = b g = fx 2 R : Ax ⊇ bg: e t Definition 2.5.[1] A fuzzy vector X = (~x1; : : :; x~n) , given byx ~i(r) = [xi(r); xi(r)], for 1 ≤ i ≤ n and 0 ≤ r ≤ 1, is called the minimal symmetric solution of a fully fuzzy system AX = b which is placed in the CSS if, for any arbitrary symmetric solution e t e e Y = (~y1; : : :; y~n) which is placed in the CSS, that is, Y (1) = X(1), we have e ⊇ e ⊇ ≥ 8 Y X; i.e.; (~yi x~i) i.e.; (σy~i σx~i ); i = 1; : : : ; n; where σy~i and σx~i are the symmetric spreads ofy ~i andx ~i, respectively. e t Definition 2.6. A fuzzy vector X = (~x1; : : :; x~n) given byx ~i(r) = [xi(r); xi(r)], for 1 ≤ i ≤ n and 0 ≤ r ≤ 1, is called the maximal symmetric solution of a fully fuzzy system AX = b which is placed in the CSS if, for any arbitrary symmetric solution e t e e Z = (~z1; : : :; z~n) which is placed in the CSS, that is, Z(1) = X(1), we have e ⊇ e ⊇ ≥ 8 X Z; i.e.; (~xi z~i) i.e.; (σx~i σz~i ); i = 1; : : : ; n; where σx~i and σz~i are the symmetric spreads ofx ~i andz ~i, respectively. 3 e t Definition 2.7. We say that a fuzzy number λ~ and a fuzzy vector X = (~x1; : : :; x~n) , given, ~ ≤ ≤ ≤ ≤ respectively, by λ(r) = [λ(r); λ(r)],x ~i(r) = [xi(r); xi(r)], for 1 i n and 0 r 1, are called a pair of minimal symmetric eigenvalue and eigenvector of Ae if, for any arbitrary e t pair of symmetric eigenvalue and eigenvectorµ ~, Y = (~y1; : : :; y~n) , withµ ~(1) = λ~(1) and Ye(1) = Xe(1), we have ⊇ ~ ≥ µ~ λ, i.e.; (σµ~ σλ~); e ⊇ e ⊇ ≥ 8 Y X; i.e.; (~yi x~i) i.e.; (σy~i σx~i ); i = 1; : : : ; n; ~ where σµ~, σλ~, σy~i and σx~i are the symmetric spreads ofµ ~, λ,y ~i andx ~i, respectively. e t Definition 2.8. We say that a fuzzy number λ~ and a fuzzy vector X = (~x1; : : :; x~n) , given, ~ ≤ ≤ ≤ ≤ respectively, by λ(r) = [λ(r); λ(r)],x ~i(r) = [xi(r); xi(r)], for 1 i n and 0 r 1, are called a pair of maximal symmetric eigenvalue and eigenvector of Ae if, for any arbitrary e t pair of symmetric eigenvalue and eigenvectorν ~, Z = (~z1; : : :; z~n) , withν ~(1) = λ~(1) and Ze(1) = Xe(1), we have ~ ⊇ ≥ λ ν;~ i.e.; (σλ~ σν~); e ⊇ e ⊇ ≥ 8 X Z; i.e.; (~xi z~i) i.e.; (σx~i σz~i ); i = 1; : : : ; n; ~ where σν~, σλ~, σx~i and σz~i are the symmetric spreads ofν ~, λ,x ~i andz ~i, respectively.