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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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