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International Journal of Innovative Technology and Exploring Engineering (IJITEE)
ISSN: 2278-3075 (Online), Volume-9 Issue-2, December 2019
Finding Inverse of a Fuzzy Matrix using Eigen
value Method
Hamed Farahani, M. J. Ebadi, Hossein Jafari
Abstract: The present paper extends a concept of the inverse of The study of the second class is ignored because of the
a matrix that its elements are fuzzy numbers, which may be complex arithmetic structure. In the present paper, the focus
implemented to model imprecise and uncertain features of the will be on this fuzzy matrices class. The investigations of
problems in the real world. The problem of inverse calculation of the invertibility of the square interval matrices and obtaining
a fuzzy matrix is converted to solving a fuzzy polynomial their inverse are two popular problems that have been warm
equations (FPEs) system. In this approach, the fuzzy system is issues in recent studies. This paper proposes an approach to
transformed to an equivalent system of crisp polynomial compute the fuzzy inverse matrix on the base of eigenvalue
equations. The solutions of the crisp polynomial equations system approach. In this method, finding the fuzzy inverse matrix is
is computed using eigenvalue method. Also, using Gröbner basis on the base of changing the fuzzy matrix into a crisp
properties a criteria for invertibility of the fuzzy matrix is
introduced. Furthermore, a novel algorithm is proposed to find a polynomial equations system. Then, a Gröbner basis
fuzzy inverse matrix. Achieving all entries of a fuzzy inverse regarding to a term order may be computed for the ideal
matrix at a time is a big advantage comparing the existence which is generated by the crisp polynomials system. The
methods. In the end, some illustrative examples are presented to Gröbner basis regarding each arbitrary term order can be
demonstrate the algorithm and concepts. computed. Moreover, the Gröbner basis computation w.r.t
Keywords : Eigenvalue, Fuzzy numbers, Fuzzy matrix, Fuzzy the lexicographical term order regarding the other terms of
identity matrix, Fuzzy linear equation system. the order is more complex in the computational complexity
I. INTRODUCTION viewpoint [1]. Hence, a suitable term order can be selected
for computing the Gröbner basis and reducing the
When the fuzzy uncertainty happens in a problem, the computations rate. In the eigenvalue approach, the roots
fuzzy matrices are successfully applied. In the last two calculation of a system is accomplished separately from
decades, fuzzy matrices have been popular [24]. In matrix each other. Accordingly, the occurred approximation and the
theory, the position of the theory of generalized inverse of a probable error in the previous roots calculation don’t
fuzzy matrix is outstanding [5, 6]. The research of influence the next roots computation. In the presented
convergence of powers of a fuzzy matrix began by approach, the inverse computation of a fuzzy matrix is
Thomasan [22] in 1977. A systematic improvement to the converted to obtaining the eigenvalues of a matrix.
fuzzy matrix theory was given by Kim and Roush [20]. Therefore, the valuable tools from linear algebra can be used
Also, they proposed the algorithms to obtain a fuzzy inverse for instance to transform a matrix into triangular one
through using the properties of determinant and elementary
matrix and its generalized inverse. The ’fuzzy matrix’ term row operations. Also, a criteria is presented on the base of
is the principal idea of the present paper and has more than Gröbner basis for invertibility of the fuzzy matrix. The
two various meanings in the research. In the first class organization of the paper is as follows. In Section 2, some
A=(a_ij )_(m×n) is said to be a fuzzy matrix, if necessary results and definitions of fuzzy numbers are
a_ij∈[0,1],(i=1,2,...,m;j=1,2,...,n). They have been first mentioned. Then, the necessary results and concepts of
defined in detail and appeared with the fuzzy relations in Gröbner basis and polynomials are given in Section 3. In
[20]. Then, there was more attention in this case [7, 18, 21]. addition, a new method for calculating the fuzzy inverse
For example, the Gödel-implication operator was used by matrix is presented in Section 3. Also, a criteria and an
Hashimoto [18] and he presented some features of sub- algorithm are presented to find the fuzzy inverse matrix
inverse of the fuzzy matrices of the first kind. Also, the when it has inverse. Section 4 containing some examples
properties of their regularity were introduced by Cho in which illustrate the algorithm. Our conclusions are
1999 [7]. Moreover, a matrix including entries of fuzzy summarized in Section 5.
numbers is known as the fuzzy matrix, too [4, 11, 12, 19].
II. FUZZY BACKGROUND
In this section some preliminaries and required
Revised Manuscript Received on December 30, 2019. background on fuzzy arithmetic, fuzzy numbers, fuzzy
* Correspondence Author matrices and notation of fuzzy set theory are given.
Hamed Farahani, Department of Mathematics, Chabahar Maritime
University, Chabahar, Iran.
*
M. J. Ebadi , Department of Mathematics, Chabahar Maritime Definition 2.1 [10, 23] A fuzzy number on the set of real
University, Chabahar, Iran. numbers is a fuzzy set if its membership function ,
Hossein Jafari, Department of Mathematics, Chabahar Maritime
University, Chabahar, Iran. is as follows:
© The Authors. Published by Blue Eyes Intelligence Engineering and
Sciences Publication (BEIESP). This is an open access article under the
CC-BY-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/
Retrieval Number: B6295129219/2019©BEIESP Published By
DOI: 10.35940/ijitee.B6295.129219 3030 Blue Eyes Intelligence Engineering
Journal Website: www.ijitee.org & Sciences Publication
Finding Inverse of a Fuzzy Matrix using Eigenvalue Method
Fuzzy matrix was defined and introduced as a rectangular
array of fuzzy numbers [10, 11]. Thus, the definition in
formal form was defined as below [9]:
Definition 2.4 If the elements of a matrix are
fuzzy numbers we say that is a fuzzy matrix. In addition,
Definition 2.2 The fuzzy number with the following consider and as two fuzzy matrices of
function of membership : orders and , respectively. The is the
product order of two fuzzy matrices and their product is
given as below:
where , where the approximated
is called a number of type where the non-increasing multiplication denoted by .
continuous functions and defined on decrease In an analogous manner, a fuzzy matrix A including the
strictly to zero in those subintervals of in which they spread parts of the left and right and the center just as a
are satisfying the conditions and positive. fuzzy number is given as the following form: ,
Also, the non-negative real numbers and are the spread in which , and represent the center, left and right
parameters. Usually, and are known as the shape spread matrices respectively and all are crisp. Moreover, the
functions. In addition, the following is parametric form of sizes of them are the same [9]. The interested reader can
fuzzy numbers of type [11]: refer to [24] for further basic and essential properties of the
, where left and right spreads are respectively fuzzy matrices.
and . The symmetric fuzzy number is a fuzzy number III. POLYNOMIALS AND GRÖBNER BASIS
in which the spreads are [11]. This section contains the introduction of some basic
The operations of arithmetic were defined by Dubois and concepts in relation to the Gröbner basis and polynomials.
Prade [11] relying on the parametric representations of the Consider as a field and as (algebraically
fuzzy numbers of type. Here, multiplication and addition independent) variables. Each power product is
are given for the purposes of illustration. All formulas and called a monomial where . Because of
more detailed descriptions can be found in [11]. The simplicity, we abbreviate such monomials by where is
addition and multiplication for two positive fuzzy numbers used for the sequence and . The
of type and are given set of all monomials can be sort over via special kinds of
respectively as follows: total orderings which known as the orderings of monomial
recalled as the below definition.
Definition 3.1 The total ordering on the monomials set is
and named orderings of monomial if for each and as
monomials we have:
• ,
It is noticeable that the outcome fuzzy number is a kind of and
approximate results. In the below, we can find the scalar
multiplication: • is well-ordering.
There are infinitely many monomial orderings, each one
is convenient for a special type of problems. Between them,
The definitions of being positive, negative, and zero of a the graded and pure reverse lexicographic orderings
fuzzy number are given below: represented by and pointed out as follows:
Assume that . We say that
Definition 2.3 When is a fuzzy number support, if
the fuzzy number considered as positive. Thus, • whenever
if the fuzzy number considered as negative. for an integer
Lastly, if the fuzzy number considered as zero. .
Retrieval Number: B6295129219/2019©BEIESP Published By
DOI: 10.35940/ijitee.B6295.129219 3031 Blue Eyes Intelligence Engineering
Journal Website: www.ijitee.org & Sciences Publication
International Journal of Innovative Technology and Exploring Engineering (IJITEE)
ISSN: 2278-3075 (Online), Volume-9 Issue-2, December 2019
• if the above polynomial system or to the ideal is defined to
be
breaking ties when there exists an integer such
that
where is used to denote the algebraic closure of . Now,
It is worth noting that the former has many theoretical consider as a Gröbner basis for w.r.t an arbitrary
importance while the latter speeds up the computations and monomial ordering. As an interesting fact, which
carries fewer information out. Any polynomial on indicates that . This is the key computational
over can be written as linear combination of trick to solve a polynomial system. Let us continue by an
monomials. The polynomial ring on over example.
represented by or just by and is the set of Example 3.4 We are going to solve the following
all polynomials having the structure of a ring with common polynomial system:
polynomial multiplication and addition. The leading
monomial of is the greatest nominal with respect to
included in and represented by . The leading
coefficient of is the coefficient of and denoted by
.Moreover, is said to be if is
a polynomials set, and is said to be the initial ideal of By the nice properties of pure lexicographical ordering, the
and the ideal generated by if is an ideal. Now, we reduced Gröbner basis of the ideal
decide to mention the idea of Gröbner basis of polynomial has
ideal which carries lots of useful information out about the the form
ideal.
Definition 3.2 Consider as a monomial ordering and as
a polynomial ideal of . The finite set is said to be with respect to , where
a Gröbner basis of if for any nonzero polynomial , 15 14 12 10 9 8
and for some , is divisible by . g1(z ) z 3z 5z 3z z z
4z66z44z21
Using the famous basis theorem of Hilbert (See [2]), it is
shown that any polynomial ideal holds a Gröbner basis w.r.t g (z )2z14 9z1311z12 2z117z10
2
any monomial ordering. There exist also some efficient
3z92z8z74z67z510z4
algorithms to calculate Gröbner basis. The first and the most
simplest one is the Buchberger algorithm which is devoted 32
6z 11z 2z 4
in the same time of introducing the Gröbner basis concept
13 12 11 10 9
while the most efficient known algorithm is the Faugère’s g3(z )z 3z z 2z z
F algorithm [15] and another signature-based algorithms z82z62z4z33z21
such as G V [16] and GVW [17]. It is worth noting that
Gröbner basis of an ideal is not necessarily unique. To have
uniquity, we define the reduced Gröbner basis concept. We This special form of Gröbner basis for this system allows us
have the uniqueness of the reduced Gröbner basis of an ideal to find by solving only one univariate polynomial
up to the monomial ordering as a significant reality. and putting the roots into the two last polynomials in
Definition 3.3 Consider as a Gröbner basis for the ideal .
w.r.t. . Then is so called a reduced Gröbner basis of Theorem 3.5 Suppose that is a reduced Gröbner basis
whenever each is monic, which means that for w.r.t any monomial ordering and is an ideal in .
and for each none of the appearing If then .
monomials in is divisible by .
The help of Gröbner basis to solve a polynomial system The existence of univariate polynomials in a polynomial
is one of its most applications. Consider ideal depends on the dimension of the ideal. The concept of
dimension of an ideal is recalled in the next definition.
Definition 3.6 Consider as a set of variables and
as an ideal. The set of variables is said an
independent set w.r.t , whenever .
as a polynomial system and as the ideal
generated by . The affine variety corresponding to
Retrieval Number: B6295129219/2019©BEIESP Published By
DOI: 10.35940/ijitee.B6295.129219 3032 Blue Eyes Intelligence Engineering
Journal Website: www.ijitee.org & Sciences Publication
Finding Inverse of a Fuzzy Matrix using Eigenvalue Method
The dimension of is the maximal independent set a basis for
cardinality w.r.t . Furthermore, when the dimension of is for do
zero called a zero dimensional ideal, and positive the eigenvalue set of
dimensional otherwise. Zero dimensional ideals have very
nice properties which facilitate the computations. For end for
instance, for an ideal with zero dimension, the dimension
of the vector space is finite and one can find its basis for do
easily via reading the leading monomials of a Gröbnr basis. if for an then
A basis for in which the set of all monomials in
denoted by and can be constructed by the set end if
end for
Return ;
More precisely, the computation of a Gröbner basis at first Using linear algebra, eigenvalue method is a simple and
is enough to compute , and carry those monomials out efficient method to solve a zero dimensional ideal.
which are not divisible by for each . A new However, the result of cartesian product of eigenvalues
property of the ideals with zero dimension in described in gives a superset of the solution set and so it is needed, as
the below theorem in which it is one of the essential mentioned in the algorithm, to check whether a tuple is a
theorems in the present paper. However, the following solution or not. For instance to solve Example 3.4 by this
definition should be given. method we need to check tuples to find out whether
they belong to the solution set or not. This is while this
Definition 3.7 Suppose that is a polynomial ideal with system has only solutions. Thus, this method is
zero dimension and is a basis for . The definition convenient when the degree of univariate is low w.r.t the
of the linear transformation for every polynomial number of variables.Now, the eigenvalue approach
is as below: illustrated to find the real solutions of a polynomial system
through the below example.
Example 4.2 [13] The following system of equations is
considered:
Also, suppose that is the matrix representation of
w.r.t . Therefore, is said the multiplication matrix of
w.r.t .
IV. PROPOSED METHODOLOGY Suppose that is the generated ideal via these equations.
At first, a Gröbner basis for w.r.t the graded reverse
4.1 Eigenvalues for solving 0-D polynomial system lexicographic order is computed and so the command
The following theorem and algorithm state the method of
using eigenvalues to solve a zero dimensional polynomial is used to find the below monomials:
system.
Theorem 4.1 ([2]) Using the above notations, the
eigenvalues of show the values of over . Now, the matrix representation of w.r.t is obtained as
As a very fast conclusion of this theorem, one can solve a follows:
zero dimensional polynomial equations system by
calculating the eigenvalues of for each variable . Of
course the eigenvalues of are the th component of
.
Algorithm 1. Eigenvalue Method
Require: a set of
polynomials where
Ensure:
a Gröbner basis for w.r.t an arbitrary
monomial ordering;
Retrieval Number: B6295129219/2019©BEIESP Published By
DOI: 10.35940/ijitee.B6295.129219 3033 Blue Eyes Intelligence Engineering
Journal Website: www.ijitee.org & Sciences Publication
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