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hindawipublishingcorporation advancesinfuzzysystems volume2012 articleid318069 9pages doi 10 1155 2012 318069 research article fuzzysymmetricsolutionsoffuzzymatrixequations 1 2 xiaobinguo anddequanshang 1college of mathematics and information science northwest normal university lanzhou 730070 china 2departmentofpublic ...

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             HindawiPublishingCorporation
             AdvancesinFuzzySystems
             Volume2012,ArticleID318069,9pages
             doi:10.1155/2012/318069
             Research Article
             FuzzySymmetricSolutionsofFuzzyMatrixEquations
                                       1                         2
                       XiaobinGuo andDequanShang
                       1College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, China
                       2DepartmentofPublic Courses, Gansu College of Chinese Medicine, Lanzhou 730000, China
                       CorrespondenceshouldbeaddressedtoXiaobinGuo,guoxb@nwnu.edu.cn
                       Received 3 April 2012; Accepted 29 April 2012
                       AcademicEditor: F. Herrera
                       Copyright © 2012 X. Guo and D. Shang. This is an open access article distributed under the Creative Commons Attribution
                       License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
                       cited.
                                                                                                                               
                       ThefuzzysymmetricsolutionoffuzzymatrixequationAX = B,inwhichAisacrispm×mnonsingularmatrixandBisanm×n
                       fuzzy numbers matrix with nonzero spreads, is investigated. The fuzzy matrix equation is converted to a fuzzy system of linear
                       equations according to the Kronecker product of matrices. From solving the fuzzy linear system, three types of fuzzy symmetric
                       solutions of the fuzzy matrix equation are derived. Finally, two examples are given to illustrate the proposed method.
             1. Introduction                                                   has a wide use in control theory and control engineering,
                                                                               few works have been done in the past decades. In 2010, Guo
             Linear systems always have important applications in many         et al. [22–24] investigated a class of fuzzy matrix equations
             branchesofscienceandengineering.Inmanyapplications,at                     
                                                                                 X = B in which A is an m × n crisp matrix and the
             least some of the parameters of the system are represented by     A
                                                                                                        
             fuzzy rather than crisp numbers. So, it is immensely impor-       right-hand side matrix B is an m × l fuzzy numbers matrix
             tant to develop a numerical procedure that would appro-           by means of the block Gaussian elimination method and
             priately treat general fuzzy linear systems and solve them.       the undetermined coefficients method, and they studied
             The concept of fuzzy numbers and arithmetic operations            least squares solutions of the inconsistent fuzzy matrix
                                                                                                
             with these numbers was first introduced and investigated by        equation Ax = B by using the generalized inverses. In 2011,
             Zadeh [1], Dubois et al. [2], and Nahmias [3]. A different         AllahviranlooandSalahshour[25]obtainedfuzzysymmetric
             approach to fuzzy numbers and the structure of fuzzy              approximate solutions of fuzzy linear systems by solving
             numberspaceswasgivenbyPuriandRalescu[4],Goetschell                a crisp system of linear equations and a fuzzified interval
             et al. [5], and Wu and Ming [6, 7].                               systemoflinearequations.Meanwhile,they[26]investigated
                 Since Friedman et al. [8, 9]proposedageneralmodel             the maximal and minimal symmetric solutions of full fuzzy
                                                                                                   
             for solving an n × n fuzzy linear systems whose coefficients        linear systems Ax = b by the same approach.
             matrix is crisp and the right-hand side is an arbitrary fuzzy         In this paper, we propose a general model for solving
                                                                                                                 
                                                                                                            X = B where A is crisp m × m
             numbers vector by an embedding approach in 1998, many             the fuzzy matrix equation A
                                                                                                        
             works have been done about how to deal with some fuzzy            nonsingular matrix and B is an m×n fuzzy numbers matrix
             linear systems with more advanced forms such as dual              with nonzero spreads. The model is proposed in this way,
             fuzzy linear systems (DFLSs), general fuzzy linear systems        that is, we first convert the fuzzy matrix equation to a fuzzy
             (GFLSs), fully fuzzy linear systems (FFLSs), dual full fuzzy      system of linear equations based on the Kronecker product
             linear systems (DFFLSs), and general dual fuzzy linear            of matrices and then obtain three types of fuzzy symmetric
             systems (GDFLSs). These works were performed mainly by            solutions of the fuzzy matrix equation by solving the fuzzy
             Allahviranloo et al. [10–13], Abbasbandy et al. [14–17],          linear systems. Finally, some examples are given to illustrate
             Wang et al. [18, 19] and Dehghan et al. [20, 21], among           our method. The structure of this paper is organized as
             others. However, for a fuzzy matrix equation which always         follows.
                   2                                                                                                                                           AdvancesinFuzzySystems
                                                                                                                                                                    m×n                         p×q
                        In Section 2, we recall the fuzzy number and present the                               Definition 4. Suppose A = (a ) ∈ R                         , B = (b ) ∈ R             ,
                                                                                                                                                         ij                          ij
                   conceptofthefuzzymatrixequationanditsfuzzysymmetric                                         the matrix in block form:
                   solutions. The method to solve the fuzzy matrix equation                                                           ⎛a BaB ··· a B⎞
                   is proposed and the fuzzy symmetric solutions of the fuzzy                                                         ⎜ 11          12               1n   ⎟
                                                                                                                                         a BaB ··· a B
                   matrix equation are obtained in detail in Section 3.Some                                                           ⎜ 21          12               2n   ⎟        mp×nq
                                                                                                                         A⊗B=⎜ .                     .        .       .   ⎟∈R                    (2)
                                                                                                                                      ⎜ .            .        .       .   ⎟
                   examples are given to illustrate our method in Section 4 and                                                       ⎝ .            .        .       .   ⎠
                   the conclusion is drawn in Section 5.                                                                                 a BaB ··· a B
                                                                                                                                          m1        m2               mn
                   2. Preliminaries                                                                            is said the Kronecker product of matrices A and B,denoted
                                                                                                               simply by A⊗B = (a B).
                                                                                                                                              ij
                   2.1. Fuzzy Numbers. There are several definitions for the                                                                                        m×n
                                                                                                               Definition 5. Let A = (a ) ∈ R                            , a     = (a ,a ,...,
                   concept of fuzzy numbers (see [1, 2, 4]).                                                                                          ij                    i           1i   2i
                                                                                                               a )T, i = 1,...,n, the mn dimensions vector:
                                                                                                                 mi
                   Definition 1. A fuzzy number is a fuzzy set like u : R → I =                                                                               ⎛a ⎞
                                                                                                                                                             ⎜ 1⎟
                   [0,1] which satisfies the following:                                                                                                          a
                                                                                                                                                             ⎜ 2⎟
                                                                                                                                                    ( )      ⎜ ⎟
                                                                                                                                               Vec A =⎜ . ⎟                                      (3)
                        (1) u is upper semicontinuous,                                                                                                       ⎝. ⎠
                                                                                                                                                                 .
                                                                                                                                                                a
                        (2) u is fuzzy convex, that is, u(λx +(1− λ)y) ≥                                                                                          n
                             min{u(x),u(y)}forall x, y ∈ R, λ ∈ [0,1],                                         is called the extension on column of the matrix A.
                        (3) u is normal, that is, there exists x0 ∈ R such that                                Lemma6. Let A = (a ) ∈ Rm×n, B = (b ) ∈ Rn×s,and
                             u(x0) = 1,                                                                                                         ij                            ij
                                                                                                                                  s×t
                        (4) suppu ={x ∈ R | u(x) > 0} is the support of the u,                                 C=(cij)∈R .Then,                                    

                             andits closure cl(suppu)iscompact.                                                                          (        )        T               ( )
                                                                                                                                    Vec ABC = C ⊗A Vec B .                                       (4)
                        Let E1 be the set of all fuzzy numbers on R.                                                                              
                                                                                                               Definition 7. AmatrixA = (a ) is called a fuzzy matrix,
                                                                                                                                                             ij
                                                                                                                                                 
                                                                                                               if each element a           of A is a fuzzy number, that is, a                    =
                   Definition 2. A fuzzy number u in parametric form is a pair                                                            ij                                                    ij
                                                                                                               (a (r),a (r)), 1 ≤ i ≤ m,1≤ j ≤ n,0≤ r ≤ 1.
                   (u,u) of functions u(r), u(r), 0 ≤ r ≤ 1, which satisfies the                                   ij       ij
                   requirements:
                                                                                                                                                                                      m×n
                                                                                                               Definition 8. Let A = (a                = (a (r),a (r)) ∈ E                  , a   =
                                                                                                                                                    ij        ij        ij                      i
                        (1) u(r) is a bounded monotonic increasing left continu-                                                        T
                                                                                                               (a   , a , ..., a   ) , j = 1,...,n. Then, the mn dimensions
                             ousfunction,                                                                         1j    2j         mj
                                                                                                               fuzzy numbers vector:
                        (2) u(r) is a bounded monotonic decreasing left contin-                                                                               ⎛ ⎞
                                                                                                                                                                a
                             uousfunction,                                                                                                                    ⎜ 1⎟
                                                                                                                                                     
         a
                                                                                                                                                              ⎜ 2⎟
                        (3) u(r) ≤ u(r), 0 ≤ r ≤ 1.                                                                                                          ⎜ ⎟
                                                                                                                                               Vec A =⎜ . ⎟                                      (5)
                                                                                                                                                              ⎝. ⎠
                                                                                                                                                                  .
                        Acrispnumberxissimplyrepresentedby(u(r),u(r)) =                                                                                         a
                                                                                                                                                                  n
                   (x,x), 0 ≤ r ≤ 1. By appropriate definitions the fuzzy
                                                                                                       1                                                                                    
                   number space {(u(r),u(r))} becomes a convex cone E                                          is called the extension on column of the fuzzy matrix A.
                   which could be embedded isomorphically and isometrically
                   into a Banach space.                                                                        2.3. Fuzzy Matrix Equations
                   Definition 3. Let x = (x(r),x(r)), y = (y(r), y(r)) ∈ E1,                                    Definition 9. The matrix system:
                   0 ≤ r ≤ 1, and real number k ∈ R.Then,                                                                 ⎛a        a      ··· a ⎞⎛x                x     ··· x ⎞
                                                                                                                          ⎜ 11        12             1m⎟⎜ 11           12            1n⎟
                        (1) x = y iff x(r) = y(r)and x(r) = y(r),                                                             a      a      ··· a              x     x     ··· x
                                                                                                                          ⎜ 21        12             2m⎟⎜ 21           12            2n⎟
                                                                                                                          ⎜ .         .      .       .   ⎟⎜ .          .      .       .  ⎟
                                                                                                                          ⎜ .         .      .       .   ⎟⎜ .          .      .       .  ⎟
                        (2) x + y = (x(r)+y(r), x(r)+y(r)),                                                               ⎝ .         .      .       .   ⎠⎝ .          .      .       .  ⎠
                                                                                                                             a      a      ··· a              x     x     ··· x
                        (3) x − y = (x(r)− y(r), x(r)− y(r)),                                                                  m1     m2            mm         m1     m2             mn
                        (4)                                                                                                           ⎛                       ⎞                               (6)
                                                                                                                                         b      b      ··· b
                                                                                                                                      ⎜ 11        12            1n⎟
                                                ⎧                                                                                     ⎜                       ⎟
                                                                                                                                         b      b      ··· b
                                                ⎨                                                                                     ⎜ 21        12            2n⎟
                                                   (   ( )       ( ))
                                                    kx r ,kx r ,           k ≥ 0,                                                  =⎜ .           .      .       .  ⎟,
                                        kx=                                                         (1)                               ⎜ .         .      .       .  ⎟
                                                ⎩                                                                                     ⎝ .         .      .       .  ⎠
                                                   (   ( )       ( ))
                                                    kx r ,kx r ,           k<0.                                                                             
                                                                                                                                        b       b      ··· b
                                                                                                                                          m1     m2             mn
                                                                                                                                                                                    
                   2.2.   Kronecker Product of Matrices and Fuzzy Matrix. The                                  where a ,1≤ i, j ≤ m are crisp numbers and b ,1≤ i ≤
                                                                                                                          ij                                                          ij
                   following definitions and results about the Kronecker prod-                                  m,1≤ j ≤ n are fuzzy numbers, is called a fuzzy matrix
                   uct of matrices are from [27].                                                              equations (FMEs).
                    AdvancesinFuzzySystems                                                                                                                                                                      3
                          Usingmatrixnotation, we have                                                                wehave
                                                                                                         (7)                                    ⎛                                   ⎞
                                                           AX =B.                                                                                    b AbA ··· b A
                                                                                                                                                  ⎜ 11          22               l1   ⎟
                                                                                                                                          
         b AbA ··· b A                              

                          Afuzzynumbersmatrix:                                                                                                    ⎜ 12          22               l2   ⎟
                                                                                                                                                 ⎜                                   ⎟          
                                                                                                                             Vec AXB =⎜ .                        .        .       .   ⎟Vec X
                                                     
                         
                                                                ⎝ .            .        .       .   ⎠
                                                                                                                                                       .        .        .       .                         (13)
                                                                   ( )       ( )
                                         X = x          = x r ,x r ,
                                                    ij           ij       ij                                                                         b AbA ··· b A
                                                                                                           (8)                                         1l       2l                ls
                                         1 ≤ i ≤ m,1≤ j ≤ n,0≤r ≤1,                                                                                T                

                                                                                                                                               = B ⊗A Vec X .
                    is called a solution of the fuzzy linear matrix equation (6)if
                     
                    Xsatisfies                                                                                                                                         m×n
                                                                                                                      Theorem11. ThematrixX ∈E                               is the solution of the fuzzy
                                                                                                         (9)                                                                                      mn
                                                           AX =B.                                                     matrix equation (7) ifandonlyifx = Vec(X) ∈ E                                      is the
                    Clearly, Definition 9 is just for the fuzzy matrix equation                                        solution of the following linear fuzzy system:
                    and its exact solution. In this paper we will discuss its                                                                                 Gx = y,                                     (14)
                    approximate fuzzy symmetric solutions.
                                                                                                                                                                      
                                                                                                                      whereG=I ⊗Aandy=Vec(B).
                    3. MethodforSolvingFMEs                                                                                             n
                    In this section, we will investigate the fuzzy matrix equation                                    Proof. Setting B = In in (10), we have
                    (7), that is, convert it to a crisp system of linear equations and                                                              
          (          )      

                    a fuzzified interval system of linear equations, define three                                                               Vec AX = In⊗A Vec X .                                         (15)
                    types of fuzzy approximate symmetric solution and give its                                              ApplyingtheextensionoperationtheDefinition8totwo
                    solutionrepresentationtotheoriginalfuzzymatrixequation.                                           sides of (7), we also have
                          At first, we convert the fuzzy matrix equation (7)toa
                    fuzzy system of linear equations based on the Kronecker                                                                                   Gx = y,                                     (16)
                    product of matrices.
                                                                                m×n                                                                                                                   
                                                                                                                      whereG=I ⊗Aisanmn×mnmatrixandy=Vec(B)isan
                    Theorem10. Let A = (a ) belong to R                               ,letX = (x ) =                                    n
                                                           ij                                           ij
                                                          n×l                                              l×s                                                                 
                    (x (r),x (r)) belong to E                  ,andletB = (b ) belong to R                     .      mnfuzzy numbers vector. Thus, the X is the solution of (7)
                       ij        ij                                                   ij
                    Then,                                                                                                                                                     
                                                       
                  
       
                                whichisequivalenttothatx = Vec(X) which is the solution
                                                                  T                                                 of (14).
                                         Vec AXB = B ⊗A Vec X .                                          (10)               For simplicity, we denote p = mn in (7), thus
                                                                                                           m                                          ⎛                              ⎞
                    Proof. Let X = (x ,x ,...,x ), x                  = (x (r),x (r)) ∈ E ,
                                                 1   2          n      j          ij        ij                                                            g      g      ··· g
                    i = 1,2,...,m, j = 1,2,...,l. B = (b ,b ,...,b ), b ∈ Rn,                                                                          ⎜ 11        12             1,p⎟
                                                                              1    2          l     j                                                     g      g      ··· g
                    j = 1,2,...,l.Then,                                                                                                                ⎜ 21        12             2,p⎟
                                                                                                                                               G=⎜ .               .       .       .  ⎟,
                                                                                                                                                       ⎜ .         .       .       .  ⎟
                                                                                             ⎛  ⎞                                                     ⎝ .         .       .       .  ⎠
                                                                                                AXb
                                                                                             ⎜         1⎟                                                gp,1 gp,2 ··· gp,p
                                                                                             ⎜  ⎟                                                                                                          (17)
                                                                                             ⎜          ⎟                                                          ⎛ ⎞
                                                                                                AXb
                                      
                                              
     ⎜         2⎟                                                             y
                                                                                         ⎜          ⎟                                                          ⎜ 1⎟
                                                                                             ⎜          ⎟
                        Vec AXB =Vec AXb ,AXb ,...,AXb =                                                   .
                                                            1         2               l             .                                                                 y
                                                                                             ⎜ . ⎟                                                                 ⎜ 2⎟
                                                                                             ⎜ . ⎟                                                           y = ⎜ . ⎟
                                                                                             ⎜          ⎟                                                          ⎜ . ⎟
                                                                                             ⎝  ⎠                                                                 ⎝ . ⎠
                                                                                                AXb
                                                                                                       l                                                              y
                                                                                                                                                                        p
                                                                                                         (11)
                          Since                                                                                       in (14).
                                                                                                    ⎛b ⎞                    Thefollowingdefinitionsshowwhatthefuzzysymmetric
                                                                                                    ⎜ 1j⎟
                                                                                                    ⎜b ⎟              solutions of the fuzzy matrix equation are.
                                                                                                    ⎜ 2j⎟
                                                                                                    ⎜ ⎟
                                                                                                   ⎜ ⎟
                                   (                          )         (                          )
                      AXb = Ax ,Ax ,...,Ax b = Ax ,Ax ,...,Ax                                      .
                             j          1       2            n    j          1       2            l ⎜ . ⎟             Definition 12 (see [28]). The united solution set (USS), the
                                                                                                    ⎜ . ⎟
                                                                                                    ⎜ ⎟               tolerable solution set (TSS), and the controllable solution set
                                                                                                    ⎝blj⎠             (CSS)forthesystem(14)are,respectively,asfollows:
                                                                                                                                                                  p                    
                                                                                                                                              X = x∈R :Gx∩y=φ ,
                               =b Ax +b Ax +···+b Ax                                                                                         ∃∃                                 /
                                    1j     1       2j     2               lj     l
                                                                  
       
                                                                                        p               
                                                                                                                                               X∀∃ = x ∈R :Gx⊆ y ,                                         (18)
                               = b A,b A,...,b A Vec X ,
                                      1j       2j            nj                                                                                                                      
                                                                                                         (12)                                   X∃∀ = x ∈Rp :Gx ⊇ y .
               4                                                                                                                 AdvancesinFuzzySystems
               Definition 13. A fuzzy vector x = (x ,x ,...,x )⊺ given by                   Now, one solves the crisp linear system (21) to obtain
                                                           1   2       p
               x = [x (r),x (r)], 1 ≤ i ≤ p,0≤ r ≤ 1 is called the minimal                x , i = 1,2,..., p, that is, existed uniquely since det(G) =
                 i      i     i                                                             i
               symmetric solution of the fuzzy matrix equation (7)which                    det(I ⊗A) = det(A)n=0andsolvetheintervalequations(22)
                                                                                                n                   /
               is placed in CSS if for any arbitrary symmetric solution z =               to obtain α (r), i = 1,2,..., p.
                               ⊺                                                                       i
               (z , z , ..., z ) , which is placed in CSS, that is, x(1) = z(1),          So, without loss of generality and for simplicity to express
                  1  2       p
               wehave                                                                      the theory, it is assumed that the coefficients matrix G is
                                                                                           positive. Then, ith equation of interval system (22) is
                (      )           (       )
                 z ⊇ x , that is, z ⊇ x , that is, σ   ≥σ ,      i = 1,2,..., p,
                                     i    i              z    x
                                                          i     i                               (        ( )         ( ))
                                                                                             g   x −α r ,x +α r +···
                                                                                (19)          i1  1     i      1    i
                                                                                                                                 
                 
      (23)
                                                                                                                  ( )          ( )         ( )    ( )
                                                                                                   +g x −α r ,x +α r = y r ,y r ,
               where σ       and σ     aresymmetricspreadsofz and x,                                 ip   p     i     p     i           i      i
                         z         x                                    i         i
                          i          i
               respectively.                                                               it can be rewritten in parametric form:
                                                                         ⊺                          p               

               Definition 14. A fuzzy vector x = (x ,x ,...,x )         given by                
                                                           1   2       p                                          ( )        ( )
                                                                                                      g   x −α r       =y r ,       i = 1,2,..., p,         (24)
               x  = [x (r),x (r)], 1 ≤ i ≤ p,0≤ r ≤ 1 is called the                                   ij   j    i          i
                 i       i      i                                                                  j=1
               maximal symmetric solution of the fuzzy matrix equation
               (7) which is placed in TSS if for any arbitrary symmetric                              p                

                                                ⊺                                                                  ( )        ( )
               solution z = (z ,z ,...,z ) , which is, placed in TSS, that                          g    x +α r       =y r ,      i = 1,2,..., p.       (25)
                                  1   2       p                                                           ij  j     i         i
               is x(1) = z(1), we have                                                             j=1
                                                                                               So,aftersomecomputationsandreplacingα (r)withα (r)
                (      )           (       )                                                                                                    i           i1
                 x ⊇ z , that is, x ⊇ z , that is, σ   ≥σ ,      i = 1,2,..., p,
                                     i    i              x    z
                                                          i     i                          in (24) and replacing α (r) with α (r) in (25), (24),and(25),
                                                                                (20)                                 i           i2
                                                                                           they are transformed, respectively, to
                                                                                                                                        

               where σ       and σ     aresymmetricspreadsofz and x,                       α (r) = f     x ,...,x ,g ,...,g , y (r) ,        i = 1,2,..., p,
                         z         x                                    i         i         i1        1   1       p   i1      ip
                          i          i                                                                                               i
               respectively.                                                                                                            

                    Secondly, in order to solve the fuzzy matrix equation (7),                  ( )                                   ( )
                                                                                             α r = f x ,...,x ,g ,...,g ,y r ,                 i = 1,2,..., p.
                                                                                              i2        2   1       p   i1      ip   i
               weneedtoconsiderthefuzzysystemoflinearequation(14).                                                                                          (26)
               For the fuzzy linear system (14), we can extend it into to
                                                                                           However, α (r) is function of x ,...,x ,g ,...,g , y (r),
               a crisp system of linear equations and a fuzzified interval                                i1                        1       p   i1      ip   i
               system of linear equations to obtain its fuzzy symmetric                    α (r) is function of x ,...,x ,g ,...,g , y (r) such that
                                                                                            i2                        1        p  i1       ip   i
               solutions.                                                                  α (r)andα (r)areobtainedspreadsofithequationinsystem
                                                                                            i1          i2
                                                                                           (22).Perhaps,α (r)andα (r)donotsatisfytherestofinterval
                                                                                                            i1          i2
               Theorem 15 (see [25]). The fuzzy linear system (14) can be                  equations (22). Therefore, one should determine the reasonable
               extended into a p× p crisp function system of linear equations:             spreads according to decision makers. To this end, three type of
                           ⎛                       ⎞⎛ ⎞ ⎛              ⎞                   spreads are proposed as follows:
                                                                    ( )
                             g     g    ··· g          x         y  1
                              11    12          1,p     1          1                            ( )             ( )     ( )
                                                                                             α r =min{α r ,α r },                i = 1,2,..., p,0≤ r ≤ 1,
                           ⎜                       ⎟⎜ ⎟ ⎜ ( )⎟                                 L               i1     i2
                             g     g    ··· g          x         y  1
                           ⎜ 21     12          2,p⎟⎜ 1⎟       ⎜ 2     ⎟
                           ⎜                       ⎟⎜ ⎟ ⎜              ⎟
                               .    .     .     .       .   =       .    ,      (21)
                           ⎜ .      .     .     .  ⎟⎜. ⎟       ⎜ . ⎟                            ( )              ( )     ( )
                                                                                             α r =max{α r ,α r },                i = 1,2,..., p,0≤ r ≤ 1,
                           ⎝ .      .     .     .  ⎠⎝. ⎠       ⎝ . ⎠                          U                i1      i2
                                                                    ( )
                             gp,1 gp,2 ··· gp,p        xp        yp 1                               ( )        ( )    (      )   ( )
                                                                                                  α r =λα r + 1−λ α r , i=1,2,...,p,
                                                                                                    λ          U                L
                       (        ( )          ( ))
                   g    x −α r ,x +α r +···
                     11  1     1      1     1                                                                                                       [    ]
                                                            
                  
                                                0 ≤ r ≤ 1, λ ∈ 0,1 . (27)
                                             ( )          ( )         ( )    ( )
                             +g     x −α r ,x +α r             = y r ,y r ,
                                1p    p     1      p    1            1      1
                                                                                           Hence, by such computations, the fuzzy vector solution of
                       (        ( )          ( ))
                   g    x −α r ,x +α r +···
                     21  1     1      1     1                                              system (7) under proposed spreads (27) will be as follows. For
                                            ( )          ( )
     ( )      ( )
          i = 1,2,..., p,0≤ r, λ ≤ 1:
                             +g     x −α r ,x +α r             = y r ,y r ,
                                2p    p     1      p    1            2      2                                                           

                                                                                                                         ( )         ( ) t
                                                                                                               X = x r ,...,x r          ,
                                                 .                                                               L       1           p
                                                 .
                                                 .                                                            ( )    (        ( )         ( ))
                                                                                                            x r = x −α r ,x +α r ,
                                               
                                                            i         i     L      i    L
                                ( )          ( )                                                                                        
t
                   gp1 x1 −αp r ,x1 +αp r         +···                                                                   ( )         ( )
                                                                                                               X = x r ,...,x r           ,
                                                                                                               U       1           p
                                           ( )           ( )
         ( )    ( )                                                                           (28)
                            +app xp −αp r ,xp +αp r            = y r ,y r ,                                  ( )    (         ( )          ( ))
                                                                             p                             x  r = x −α r ,x +α r ,
                                                                     p                                       i         i    U       i    U
                                                                                (22)                                                    
t
                                                                                                                         ( )         ( )
                                                                                                               X = x r ,...,x r          ,
                                                                                                                 λ       1           p
               where y (1) ∈ R, i = 1,2,..., p and α (r), i = 1,2,..., p are
                        i                                  i                                                  ( )    (        ( )         ( ))
                                                                                                            x r = x −α r ,x +α r .
               unknownspreads.                                                                               i         i     λ      i    λ
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...Hindawipublishingcorporation advancesinfuzzysystems volume articleid pages doi research article fuzzysymmetricsolutionsoffuzzymatrixequations xiaobinguo anddequanshang college of mathematics and information science northwest normal university lanzhou china departmentofpublic courses gansu chinese medicine correspondenceshouldbeaddressedtoxiaobinguo guoxb nwnu edu cn received april accepted academiceditor f herrera copyright x guo d shang this is an open access distributed under the creative commons attribution license which permits unrestricted use distribution reproduction in any medium provided original work properly cited thefuzzysymmetricsolutionoffuzzymatrixequationax b inwhichaisacrispm mnonsingularmatrixandbisanm n fuzzy numbers matrix with nonzero spreads investigated equation converted to a system linear equations according kronecker product matrices from solving three types symmetric solutions are derived finally two examples given illustrate proposed method introduction has ...

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