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Intern. J. Fuzzy Mathematical Archive
Vol. 20, No. 1, 2022, 1-16
ISSN: 2320 –3242 (P), 2320 –3250 (online)
Published on 20 May 2022
www.researchmathsci.org
DOI: http://dx.doi.org/10.22457/ijfma.v20n1a01232
1
Row-Average-Max-Norm of Fuzzy Matrix
Jhuma Das and Suman Maity
*
Department of Mathematics
Raja Narendra Lal Khan Women’s College (Autonomous)
Midnapore-721102, India
*
Corresponding author. Email: maitysuman2012@gmail.com
Received 12 April 2022; accepted 18 May 2022
Abstract. Fuzzy matrices play a vital role in handling different models in an uncertain
environment. In this paper, we have defined the row-average-max norm of fuzzy matrices.
We also investigated some properties and lemma of the row-average-max norm of the
fuzzy matrix.
Keywords: Norm of a fuzzy matrix, row-average-max norm of fuzzy matrix, properties of
row-average-max norm of fuzzy matrix
AMS Mathematics Subject Classification (2010): 94D05
1. Introduction
An analysis of linear algebra is one of the most popular and fascinating areas in the last
few decades due to its interdependency with other applied and pure areas. Measuring the
length of vectors is an essential analysis in different theoretical development aspects in
many potential applications. For this purpose, norm functions are considered on a vector
space. A norm on a real vector space is a function satisfying
1. for any non-zero .
2. for any and .
3. for any
In general, the norm signifies the measure of the size of the vector where
equation (1) requires the size to be positive, equation (2) requires the size to be scaled as
the vector is scaled, and equation (3) is known as the triangle inequality having its origin
in the notion of distance in
. The equation (2) is called the homogeneous condition and
this condition ensures that the norm of the zero vector in is; thiscondition is often
included in the definition of a norm.
A familiar example of norms on
are the
norms, where defined
by
if and
if
for any
Jhuma Das and Suman Maity
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It is worthy to remember that if one defines an
function on
as defined above
with , then it does not satisfy the triangle inequality, hence is not a norm.
Suppose the norm on a real vector space is given. In that case, numerous aspects
can be formulated, such as one can compare the norms of vectors, discussing the
convergence of the sequence of vectors, studying limits and continuity of transformations,
and considering approximation problems such as finding the nearest element in a subset or
a subspace of to a given vector. These problems arise naturally in analysis, numerical
analysis, differential equations, Markov chains etc.
The norm determines the "size" of a matrix that is necessarily related to how many
rows or columns the matrix contains. The norm of a square matrix is a non-negative real
number denoted by . There are several different ways of defining a matrix norm, but
they all share the following properties:
1.
0≥M
for any square matrix
.M
2.
0=M
iff the matrix
0=M
.
3.
MKKM |=|
for any scaler
K
.
4.
NMNM +≤+ for any square matrix
NM ,
.
5.
NMMN
≤.
Fuzzy matrix norm:
Like vector norm and matrix norm, the norm of a fuzzy matrix is also a function
[0,1])(: →
FM
n
. which satisfies the following properties
1. for any square matrix .
2. iff the fuzzy matrix .
3. for any scalar K .
4. for any two fuzzy matrices and .
5. for any fuzzy matrix and .
In this project paper, we have defined different types of norms on fuzzy matrices.
1.1. Motivation
To analyze different geometrical and analytical structures, norms on a vector space could
be employed. The choice of utilizing the norm decides the convergence of a sequence in
an infinite dimensional vector space. This phenomena leads to many interesting questions
and research in analysis and functional analysis.
In a finite-dimensional vector space , two norms
and
are said to be
equivalent if there exist two positive constants such that
for all
.
To prove the convergence concerning one norm for a given sequence is easier than the
other. In an application such as numerical analysis, one would like to use a norm that can
determine convergence efficiently. Hence, it is a good idea to know different norms.
Secondly, in some cases, a specific norm may be needed to deal with a certain problem.
For instance, if one travels in Manhattan and wants to measure the distance from a location
marked as the origin to a destination marked as on the map, one may use the
norm of , which measures the straight line distance between two points, or one
Row-Average-Max-Norm of Fuzzy Matrix
3
may need to use the
norm of , which measures the distance for a taxi cab to drive from
to . The
norm is sometimes referred to as the taxi cab norm for this reason.
In approximation theory, solutions of a problem can vary with different problems.
For example, if is a subspace of
and does not belongs to, then for
there is a unique
such that
for all , but the uniqueness
condition may fail if or . To see a concrete example let and
. Then for all we have for all
. For some problems, having a unique approximation is good, but for others it may be
better to have many so that one of them can be chosen to satisfy additional conditions.
1.2. Literature review
It is well known that matrices play a vital role in several areas including mathematics,
physics, statistics, engineering, social sciences. An ample number of methods have been
reported in several journals as well as in books. But our real-life problems including social
science, medical science, environment etc. do not always involve crisp data. Furthermore,
owing to various types of uncertainties present in our daily life problems we cannot
successfully use traditional classical matrices. Nowadays, probability, fuzzy sets,
intuitionistic fuzzy sets, vague sets and rough sets are used as mathematical tools for
dealing with uncertainties. Fuzzy matrices arise in many applications, one of which is as
adjacency matrices of fuzzy relations and fuzzy relational equations have important
applications in pattern classification and in handing fuzziness in knowledge-based systems.
First-time Fuzzy matrices were introduced by Thomason [44], where they
discussed on the convergence of powers of a fuzzy matrix. Ragab et al. [34, 35] presented
some properties of the min-max composition of fuzzy matrices. Hashimoto [18, 19] studied
the canonical form of a transitive fuzzy matrix. Hemashina et al. [20] Investigated iterates
of fuzzy circulant matrices. Powers and nilpotent conditions of matrices over a distributive
lattice are considered by Tan [43]. After that Pal, Bhowmik, Adak, Shyamal, Mondal have
done a lot of work on fuzzy, intuitionistic fuzzy, interval-valued fuzzy, etc. matrices [1-12,
26-33, 37-41].
The elements of a fuzzy matrix lie in the closed interval. Although every
matrix, in general, is not a fuzzy matrix still we can see that all fuzzy matrices. We see the
fuzzy interval, i.e., the unit interval is a subset of reals. Thus, a matrix, in general, is not a
fuzzy matrix since the unit interval is contained in the set of reals. The big question
arises when it comes to the addition of two fuzzy matrices and and getting the sum of
them to be fuzzy matrix. The answer in general is not possible for the sum of two fuzzy
matrices may turn out to be a matrix which is not a fuzzy matrix. If we add above two fuzzy
matrices and then all entries in will not lie in , hence is only just
a matrix and not a fuzzy matrix.
Henceforth, the max or min operations could be defined in the case of fuzzy
matrices. Therefore, under the max or min operation, the resultant matrix is again a fuzzy
matrix. In general, to add two matrices we use max operation.
It is evident that the product of two fuzzy matrices under usual matrix
multiplication is not a fuzzy matrix. So, we need to define a compatible operation
analogous to the product so that the product again happens to be a fuzzy matrix. However
even for this new operation if the product is to be defined we need the number of
Jhuma Das and Suman Maity
4
columns of to be equal to the number of rows of . The two types of operation are called
max-min operation and min-max operation.
In [23], we introduced max-norm and square-max norm, row and column max-
average nom of fuzzy matrices and some properties of these two norms. In this paper, we
have defined row-average-max norm with some properties
Definition 1. An
n
n
×
fuzzy matrix A is called reflexive iff
for all i=1,2,...,n. It is
called
α
-reflexive iff
for all i=1,2,...,n where
[0,1]
∈
α
. It is called weakly
reflexive iff
for all i,j=1,2,...,n. An
n
n
×
fuzzy matrix A is called irreflexive iff
for all i=1,2,...,n.
Definition 2. An
n
n
×
fuzzy matrix S is called symmetric iff
jiij
ss =
for all i,j=1,2,...,n. It
is called antisymmetric iff
n
ISS ≤
′
∧ where
n
I is the usual unit matrix.
Note that the condition
n
ISS ≤
′
∧, means that
0=
jiij
ss
∧
for all
j
i
≠
and 1≤
ii
s for
all i. So if
1=
ij
S
then
0=
ji
s
, which the crisp case.
Definition 3. An
n
n
×
fuzzy matrix N is called nilpotent iff
0=
n
N
(the zero matrix). If
0=
m
N
and
0
1
≠
−
m
N
;
nm
≤
≤
1
then N is called nilpotent of degree m. An
n
n
×
fuzzy
matrix E is called idempotent iff
E
E
=
2
. It is called transitive iff
E
E
≤
2
. It is called
compact iff
E
E
≥
2
.
Definition 4. A triangular fuzzy matrix of order
n
m
×
is defined as nmij
aA ×
)(= where
ijijijij ma
β
α
,,= is the
th
ij element of
A
, ij
m is the mean value of ij
a and ij
α
,ij
β
are
left and right spread of ij
a respectively.
2. Preliminaries
In this section different types of matrix norm and fuzzy matrix norm are discussed.
2.1. Matrix norm
Definition 5. (The maximum absolute column sum). Simply we sum the absolute values
down each column and then take the biggest answer. (A useful reminder is that "1" is a
tall, thin character and a column is a tall, thin quantity.)
P
1
= .|)|(
max
1=
1ij
n
i
nj
p
≤≤
Definition 6. The infinity norm of a square matrix is the maximum of the absolute row sum.
Simply we sum the absolute values along each row and then take the biggest answer. The
infinity norm of a matrix A is defined by
∞
P
= .|)|(
max
1=
1ij
n
j
ni
p
≤≤
Row-Average-Max-Norm of Fuzzy Matrix
5
Definition 7. The Euclidean norm of a square matrix is the square root of the sum of all
the squares of the elements. This is similar to ordinary "Pythagorean" length where the
size of a vector is found by taking the square root of the sum of the squares of all the
elements. The Euclidean norm of a matrix A is defined by
.)(= 2
1=1= ij
n
j
n
i
EpP
Any definition you can define of which satisfies the five condition mentioned at the
beginning of this section is a definition of a norm. There are many many possibilities, but
the three given above are among the most commonly used.
2.2. Norm of fuzzy matrix
Definition 8. Max norm (Maity [23]): Max norm of a fuzzy matrix )(FMA
n
∈ is denoted
by
M
A
which gives the maximum element of the fuzzy matrix and it is defined by
M
A
=
ij
n
ji
a
∨
1=,
Definition 9. (Maity [23]): Square-max norm of a fuzzy matrix A is denoted by
SM
A
and define by
SM
A=
2
1=,
)(
ij
n
ji
a
∨=
.)( 2
M
A
In this norm at first we will find the maximum element of the fuzzy matrix and then square
it.
Definition 10. Row-max-average Norm (Maity [24]): Row-max-average norm of a fuzzy
matrix A is denoted by
RMA
A
and define by
)(
1
=
1=
1= ij
n
j
n
i
RMA
a
n
A
∨
Here, at first we find maximum element in each row. Then we determine the average of the
maximum element.
Definition 11. Column-max-average norm (Maity [24]): The Column-max-average norm
of a fuzzy matrix
A
is denoted by
CMA
A
and define by
).(
1
=
1=
1= ij
n
i
n
j
CMA
a
n
A
∨
Here we find maximum element in each column and then average of the maximum elements.
Definition 12. Pseudo norm on fuzzy matrix (Maity [24]): A norm of a fuzzy matrix is
called pseudo norm of a fuzzy matrix if it fulfill the following conditions
1.
0≥A
for any fuzzy matrix
A
.
2. if
0=A
then
0=A
.
3.
AkkA |=| for any scaler
[0,1]
∈
k
.
Jhuma Das and Suman Maity
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4.
BABA +≤+ for any two fuzzy matrix
A
and
B
.
5.
BAAB ≤ for any two fuzzy matrix
A
and
B
.
Definition 13. Max-min Norm (Maity [24]): Max-Min norm of a fuzzy matrix A is denoted
by
MM
A
and define by )(=
1=1= ij
n
j
n
i
MM
aA ∨∧
Here, first we find the maximum element in each row and then minimum of the maximum
elements.
Definition 13. Column-average-max Norm (Samanta and Maity [36]): Colum-average-
max norm of a fuzzy matrix A is denoted by
and defined by
.
Here, firstly we find the average value in each row and then find maximum of these average
values.
2.3. Addition and multiplication of fuzzy matrices
We have used the operator for addition of fuzzy matrices and used the operator for
multiplication of fuzzy matrices. This two operators are define by the following way.
If
and
Then
and
In this type of multiplication, fuzzy matrices will be of same order.
Example 1. If
and
Then
and
3. Row-average-max-norm (RAM)
Here we will define a new type of norm called Row-Average-Max norm. We will use new
type of operators of fuzzy matrices for this norm. Here, at first, we will determine the
average of the elements in each row. Then we will find the maximum element of this
Row-Average-Max-Norm of Fuzzy Matrix
7
average. Row-Average-Max Norm of a fuzzy matrix is denoted by
and defined
by
Lemma 1. All the conditions of norm are satisfied by
Proof: Let us consider
and
and
(i) As all
so according to the definition of Row-Average-Max norm
obviously
Now
for all
for all
for all
So,
iff
(ii) Here we define a new type of scalar multiplication as follows
So, if
Therefore
for all
(iii) Now,
Then
Jhuma Das and Suman Maity
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So,
(iv)
Now,
Therefore
Hence, all the conditions of norm are satisfied by Row-Average-Max.
3.1 Properties of row-average-max norm
Properties 1. If and are two fuzzy matrices then
Proof: Let us consider
and
Then
and
So,
Example 2.
Let
and
Row-Average-Max-Norm of Fuzzy Matrix
9
Then
Now,
and
Here,
,
Then
So,
Properties 2. If and are two fuzzy matrices and then
Proof: As so
for all .
This implies,
for all .
for all .
Example 3.
Let
and
Therefore, if then
Properties 3. If , and are three fuzzy matrices and
Proof: Let us consider
,
and
Then
and
Jhuma Das and Suman Maity
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and
Now,
.
So, if then
Example 4.
Let
,
and
Now,
and
Hence if , then
Properties 4. If and are two fuzzy matrices, then
Proof: Let us consider
and
Then
Now,
Hence
Row-Average-Max-Norm of Fuzzy Matrix
11
Example 5. Let
and
Now,
and
Then,
So,
Properties 5. If and are two fuzzy matrices, then
.
Proof: Let us consider
and
Then
Now,
]
So,
.
Example 6. Let
and
Now,
and
Then
and
So,
Definition 14. Define a mapping
as
for all in
.
Proposition 1. The above mapping d satisfies the following condition for all in
(i) and iff .
(ii)
(iii) for all in
Proof: (i)
. [by first condition of norm]
Again,
Jhuma Das and Suman Maity
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(ii)
[ by properties 4 ]
(iii)
=
So,
.
Example 7. Let
,
and
Now,
. Then
and
Then
and
(i)
(ii)
. Then
So, .
(iii) Here
So,
Definition 15. Define a mapping:
as
for all in
Proposition 2. The above mapping
satisfies the following condition for all
in
.
(i)
and
iff or or both .
(ii)
.
Proof: (i)
as
and
.
Now,
or
or both
either or or both
So,
.
Proposition 3. If
and then
for all
.
Row-Average-Max-Norm of Fuzzy Matrix
13
Proof: Since,, so
.
Now,
and
.
Case-1:
If
then
That is,
Case-2:
If
then
.
-3:
If
then
and
o,
.
Therefore,
for all
.
3.2. Algorithm
Input: A fuzzy matrix of order
Output: A real number in [0,1]
Max=0;
for i=1 to n
ans=0;
for j=1 to n
ans=ans+a[i][j];
end for
ans =ans/n;
if max<ans then max=ans;
end for
return max;
4. Conclusion
In this paper, two types of operators on fuzzy matrices have been used for our further
analysis. By employing these, we have defined row-average-max norm with some
important properties. The coffee industry of India is the sixth largest producer of coffee in
the world. India coffee is said to be the finest coffee grown in the shade rather than direct
sunlight anywhere in the world. Coffee cultivators in Kodai Hills faced different types of
issues including Labour shortage, Storage problem, Low margin of profit, Monsoon
failures and unseasonal rains etc. Fuzzy matrices are used to analysis the problems
encountered by the coffee cultivators in Kodai Hills. Hence, in these aspects norm of fuzzy
matrices will help in this case.
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