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Abstract

In this paper, we define two new type of operators of fuzzy matrices denoted by the symbol Å and Ä.Using these operators of fuzzy matrices we define row-maxaverage norm, column-max-average norm. Here instead of addition of fuzzy matrices we use the operator Å and instead of multiplication of fuzzy matrices we use the operatorÄ.We also define Pseudo norm of fuzzy matrices and max-min norm.
Progress in Nonlinear Dynamics and Chaos
Vol. 3, No. 1, 2015, 25-39
ISSN: 2321 – 9238 (online)
Published on 20 August 2015
www.researchmathsci.org
25
Progress in
Row and Column-Max-Average Norm and Max-Min
Norm of Fuzzy Matrices
Suman Maity
Department of Applied Mathematics with Oceanology and Compute Programming
Vidyasagar University, Midnapore - 721102, India
email: maitysuman2012@gmail.com
Received 12 July 2015; accepted 16 August 2015
Abstract. In this paper, we define two new type of operators of fuzzy matrices denoted by
the symbol
and
.
Using these operators of fuzzy matrices we define row-max-
average norm, column-max-average norm. Here instead of addition of fuzzy matrices we
use the operator
and instead of multiplication of fuzzy matrices we use the
operator
.
We also define Pseudo norm of fuzzy matrices and max-min norm.
Keywords: fuzzy matrices, row-max-average norm, column-max-average norm, pseudo
norm of fuzzy matrices, max-min norm.
1. Introduction
The study of linear algebra has become more and more popular in the last few decades.
People are attracted to this subject because of its beauty and its connection with many
other pure and applied areas. In theoretical development of the subject as well as in many
application, one often needs to measure the length of vectors. For this purpose, norm
functions are consider on a vector space.
A norm on a real vector space V is a function satisfying
1.
0>u
for any nonzero
.Vu
2.
urru |=|
for any
Rr
and
u
V.
3.
vuvu ++
for any
., Vvu
The norm is a measure of the size of the vector
u
where condition (1) requires the size to
be positive, condition (2) requires the size to be scaled as the vector is scaled, and
condition (3) is known as the triangle inequality and has its origin in the notion of
distance in
3
R
. The condition (2) is called homogeneous condition and this condition
ensure that the norm of the zero vector in V is 0; this condition is often included in the
definition of a norm.
Common example of norms on
n
R
are the
p
l
norms,where
p1
, defined by
RV :.
Suman Maity
26
p
p
j
n
j
p
uul
1
1=
}||{=)(
if
<1 p
and
)(ul
p
=max
||
1jnj
u
if
=
p
for any
.),....,,(=
21 nt
n
Ruuuu
Note that if one define an
p
l
function on
n
R
as define
above with
1<<0 p
, then it does not satisfy the triangle inequality, hence is not a norm.
Given a norm on a real vector space V, one can compare the norms of vectors, discuss
convergence of sequence of vectors, study limits and continuity of transformations, and
consider approximation problems such as finding the nearest element in a subset or a
subspace of V to a given vector. These problems arise naturally in analysis, numerical
analysis, differential equations, Markov chains, etc.
The norm of a matrix is a measure of how large its elements are. It is a way of
determining the "size" of a matrix that is necessarily related to how many rows or
columns the matrix has. The norm of a square matrix A is a non negative real number
denoted by
A
. There are several different ways of defining a matrix norm but they all
share the following properties:
1.
0A
for any square matrix
.
2.
0=A
iff the matrix
0=A
.
3.
AKKA |=|
for any scaler
K
.
4.
BABA ++
for any square matrix
BA,
.
5.
BAAB
Different types of matrix norm:
The 1-norm
A
1
=
|)|(
max
1=
1ij
n
i
nj
a
The infinity norm
A
=
|)|(
max
1=
1ij
n
j
ni
a
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.
Euclidean norm
2
1=1= )(= ij
n
j
n
i
EaA
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.
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
Row and Column-Max-Average Norm and Max-Min Norm of Fuzzy Matrices
27
the three given above are among the most commonly used.
Like vector norm and matrix norm, norm of a fuzzy matrix is also a function
[0,1])(: FM
n
.
which satisfies the following properties
1.
0A
for any fuzzy matrix
.
2.
0=A
iff the fuzzy matrix
=0.
3.
AKKA |=|
for any scaler
[0,1]
K
.
4.
BABA ++
for any two fuzzy matrix
and
B
.
5.
BAAB
for any two fuzzy matrix
and
B
.
In this project paper we will define different type of norm on fuzzy matrices.
2. Fuzzy matrix
We know that matrices play an important role in various areas such as mathematics,
physics, statistics, engineering, social sciences and many others. Several works on
classical matrices are available in different journals even in books also. But in our real
life problems in social science, medical science, environment etc. do not always involve
crisp data. Consequently, we can not successfully use traditional classical matrices
because of various types of uncertainties present in our daily life problems. Nowa days
probability, fuzzy sets, intuitionistic fuzzy sets, vague sets, rough sets are used as
mathematical tools for dealing uncertainties. Fuzzy matrices arise in many application,
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.
Fuzzy matrices were introduce for the first time by Thomason [42], who
discussed the convergence of powers of fuzzy matrix. Ragab et al. [33,34] 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 consider by Tan [41]. After that Pal, Bhowmik, Adak, Shyamal,
Mondal have done lot of works on fuzzy, intuitionistic fuzzy, interval-valued fuzzy, etc.
matrices [1-12,25-32,35-39].
The elements of a fuzzy matrix having values in the closed interval [0,1]. We
can still see that all fuzzy matrices are matrices but every matrix in general is not a fuzzy
matrix. 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 [0,1] is contained in the set of reals.
The big question is can we add two fuzzy matrices A and B and get 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
matrix A and B then all entries in A+B will not lie in [0,1], hence A+B is only just a
matrix and not a fuzzy matrix.
So only in case of fuzzy matrices the max or min operation are defined. Clearly
under the max or min operation the resultant matrix is again a fuzzy matrix. In general to
add two matrix we use max operation.
We see the product of two fuzzy matrices under usual matrix multiplication is not
Suman Maity
28
a fuzzy matrix. So we need to define a compatible operation analogous to product so that
the product again happens to be a fuzzy matrix. However even for this new operation if
the product XY is to be defined we need the number of columns of X is equal to the
number of rows of Y. The two types of operation which we can have are max-min
operation and min-max operation.
In [23], we introduced max-norm and square-max norm of fuzzy matrices and
some properties of this two norm.
In this paper, we we have introduced two new operators on fuzzu matrices
denoted by the symbol
and
. Using these operators we define different types of
norm of fuzzy matrices.
Definition 1. [41] A fuzzy matrix (FM) of order $m\times n$ is defined as  

, 

where

is the membership value of the ij-th element

in A.
An
n
n
×
fuzzy matrix R is called reflexive iff
1=
ii
r
for all i=1,2,...,n. It is called
α
-
reflexive iff
α
ii
r
for all i=1,2,...,n where
[0,1]
α
. It is called weakly reflexive iff
ijii
rr
for all i,j=1,2,...,n. An
n
n
×
fuzzy matrix R is called irreflexive iff
0=
ii
r
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
.
3. New opertors of fuzzy matrices
We already discussed addition and multiplication of fuzzy matrices in introduction. We
used max operation to add fuzzy matrices and min-max operation to multiply fuzzy
matrices till now. But here we will define new type of operators of fuzzy matrices
denoted by the symbol
and
. Instead of addition of fuzzy matrices we will use the
operator
and instead of multiplication we will use the operator
. This two new
operators are define by the following way.
If
nnnn
n
n
aaa
aaa
aaa
A
.....
.....
.....
=
21
22221
11211
and
nnnn
n
n
bbb
bbb
bbb
B
.....
.....
.....
=
21
22221
11211
Row and Column-Max-Average Norm and Max-Min Norm of Fuzzy Matrices
29
then
+++
+++
+++
2
.....
22
2
.....
22
2
.....
22
=
2211
22
22222121
1112121111
nnnnnnnn
nn
nn
bababa
bababa
bababa
BA
and
},{.....},{},{
},{.....},{},{
},{.....},{},{
=
2211
2222222121
1112121111
nnnnnnnn
nn
nn
bababa
bababa
bababa
BA
.
Must be remember that in this type of multiplication, fuzzy matrices will be of same
order.
Proposition 1. [23] ),(),(),( 21212211 bbaababa
+++
4. Row-max-average Norm
Here we will define a new type of norm called Row-Max-Average norm. We will used
new type of operators of fuzzy matrices for this norm. Here, at first we will find
maximum element in each row. Then we will determine the average of the maximum
element. 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
Lemma 1. All the conditions of norm are satisfied by
)(
1
=
1=
1= ij
n
j
n
i
RMA
a
n
A
.
Proof: Let us consider
nnnn
n
n
aaa
aaa
aaa
A
....
....
....
=
21
22221
11211
and
nnnn
n
n
bbb
bbb
bbb
B
....
....
....
=
21
22221
11211
)(
1
=
1=
1= ij
n
j
n
i
RMA
a
n
A
and
)(
1
=
1=
1= ij
n
j
n
i
RMA
b
n
B
(i) As all
0
ij
a
so according to the definition of Row-max-average norm obviously
0
RMA
A
.
Suman Maity
30
Now
0=
RMA
A
0=)(
1
1=
1= ij
n
j
n
i
a
n
0=
1= ij
n
j
a
for all i=1,2,...,n.
0=...== 21 inii aaa for all i=1,2,...,n.
0=
ij
a
for all
..,.1,2,.=, nji
0=A
So,
0=
RMA
A
iff
0=A
.
(ii) Here we define a new type of scaler multiplication as follows
RMARMA
RMA
ij
AifA
Aif
a|>|
||||
=
α
αα
α
So, if
RMA
A
||
α
then
RMARMA
AA ||=||=
ααα
and if
RMA
A|>|
α
then
.||==
RMARMARMA
AAA
αα
Therefore
RMARMA
AA |=|
αα
for all
[0,1].
α
(iii)
+++
+
++
+++
2
...
22
2
...
22
2
...
22
=
2211
2222222121
11
12121111
nnnnnnnn
nn
nn
bababa
ba
baba
bababa
BA
RMA
BA
n
bababa
nini
n
i
ii
n
i
ii
n
i
)
2
(......)
2
()
2
(
=
1=
22
1=
11
1=
+
++
+
+
+
n
bababa
ni
n
i
ni
n
i
i
n
i
i
n
i
i
n
i
i
n
i
2
)(....)()(
1=1=
2
1=
2
1=
1
1=
1
1=
++++++
n
ba
ij
n
j
n
i
ij
n
j
n
i
2
)()(
=
1=
1=
1=
1=
+
2
)(
1
)(
1
=
1=
1=
1=
1= ij
n
j
n
i
ij
n
j
n
i
b
n
a
n
+
2
=
RMARMA
BA +
RMARMA
BA =
So,
RMARMARMA
BABA
.
Row and Column-Max-Average Norm and Max-Min Norm of Fuzzy Matrices
31
(iv)
},{...},{},{
},{...},{},{
},{...},{},{
=
2211
2222222121
1112121111
nnnnnnnn
nn
nn
bababa
bababa
bababa
BA
Now
ijijij
aba },{
and
ij
b
for all
j
i
,
.
ij
n
j
ijij
n
j
aba
1=1=
)},({
and
ij
n
j
b
1=
for all
.
i
)(
1
)}],({[
1
1=
1=
1=
1= ij
n
j
n
i
ijij
n
j
n
i
a
n
ba
n
and
)(
1
1=
1= ij
n
j
n
i
a
n
RMARMARMA
BABA
Hence all the conditions of norm are satisfied by Row-max-average.
5. Properties of row-max-average Norm
Properties 1. If
A
and
B
are two fuzzy matrices then
.)(
RMA
T
RMA
T
RMA
T
BABA
Proof: Let us consider
nnnn
n
n
aaa
aaa
aaa
A
.....
.....
.....
=
21
22221
11211
and
nnnn
n
n
bbb
bbb
bbb
B
.....
.....
.....
=
21
22221
11211
.
Then
+++
+++
+
++
2
.....
22
2
.....
22
2
.....
22
=
2211
2222222121
1112121111
nnnnnnnn
nn
nn
bababa
bababa
ba
baba
BA
Suman Maity
32
and
+++
+++
+++
2
.....
22
2
.....
22
2
.....
22
=)(
2211
22
22221212
1121211111
nnnnnnnn
nn
nn
T
bababa
bababa
bababa
BA
.
)
2
(
1
=)(
1=
1=
jiji
n
j
n
i
RMA
T
ba
n
BA +
)(
2
1
1=1=
1= ji
n
j
ji
n
j
n
i
ba
n
+
2
)(
1
)(
1
1=
1=
1=
1= ji
n
j
n
i
ji
n
j
n
i
b
n
a
n
+
RMA
T
RMA
T
RMA
T
RMA
T
BA
BA
+=
2
=
Properties 2. If
A
and
B
are two fuzzy matrices and
B
A
then
RMARMA
BA
.
Proof: As
B
A
so,
ijij
ba
for all
.
,
j
i
ij
n
j
ij
n
j
ba
1=1=
for all i.
)()(
1=
1=
1=
1=
ij
n
j
n
i
ij
n
j
n
i
ba
)(
1
)(
1
1=
1=
1=
1=
ij
n
j
n
i
ij
n
j
n
i
b
n
a
n
RMARMA
BA
Properties 3. If
A
and
B
are two fuzzy matrices and
B
A
then
RMARMA
CBCA
hold.
Proof: Let us consider
nnnn
n
n
aaa
aaa
aaa
A
.....
.....
.....
=
21
22221
11211
nnnn
n
n
bbb
bbb
bbb
B
.....
.....
.....
=
21
22221
11211
and
nnnn
n
n
ccc
ccc
ccc
C
.....
.....
.....
=
21
22221
11211
.
Then
},{.....},{},{
},{.....},{},{
},{.....},{},{
=
2211
2222222121
1112121111
nnnnnnnn
nn
nn
cacaca
cacaca
cacaca
CA
Row and Column-Max-Average Norm and Max-Min Norm of Fuzzy Matrices
33
and
},{.....},{},{
},{.....},{},{
},{.....},{},{
=
2211
2222222121
1112121111
nnnnnnnn
nn
nn
cbcbcb
cbcbcb
cbcbcb
CB
.
)}],({[
1
=
1=
1= ijij
n
j
n
i
RMA
ca
n
CA
and
)}],({[
1
=
1=
1= ijij
n
j
n
i
RMA
cb
n
CB
Now
ijij
baBA for all
.
,
j
i
},{},{
ijijijij
cbca
for all
.
,
j
i
)},({)},({
1=1= ijij
n
j
ijij
n
j
cbca
for all
.
i
)}],({[
1
)}],({[
1
1=
1=
1=
1= ijij
n
j
n
i
ijij
n
j
n
i
cb
n
ca
n
RMARMA
CBCA
6. Column-max-average norm
Like Row-max-average norm we will define Column-max-average norm. Here we will
find maximum element in each column and then average of the maximum elements. Here
we will also use the new type of operators of fuzzy matrices. The Column-max-average
norm of a fuzzy matrix
A
is denoted by
CMA
A
and define by
).(
1
=
1=
1=
ij
n
j
n
i
CMA
a
n
A
Lemma 2. All the conditions of norm are satisfied by
).(
1
=
1=
1= ij
n
j
n
i
CMA
a
n
A
Proof: Let us consider
nnnn
n
n
aaa
aaa
aaa
A
...
...
...
=
21
22221
11211
and
nnnn
n
n
bbb
bbb
bbb
B
...
...
...
=
21
22221
11211
)(
1
=
1=
1= ij
n
i
n
j
CMA
a
n
A
and
)(
1
=
1=
1= ij
n
i
n
j
CMA
b
n
B
(i) As all
0
ij
a
so according to the definition of Column-max-average norm obviously
0
CMA
A
.
Now
0=
CMA
A
Suman Maity
34
0=)(
1
1=
1= ij
n
i
n
j
a
n
0=
1= ij
n
i
a
for all j=1,2,...,n.
0==...==
21 njjj
aaa for all j=1,2,...,n.
0=
ij
a
for all
.1,2,...,=, nji
.0=A
So,
0=
CMA
A
iff
0=A
.
(ii) Here we will use the same type of scaler multiplication which we used in Row-max-
average norm and that is
CMACMA
CMA
ij
AifA
Aif
a|>|
||||
=
α
αα
α
So if
CMA
A||
α
then
CMACMA
AA ||||=
ααα
=
and if
CMA
A|>|
α
then
.|=|=
CMACMACMA
AAA
αα
Therefore
CMACMA
AA |=|
αα
for all
[0,1].
α
(iii)
+++
+++
+
++
2
.....
22
2
.....
22
2
.....
22
=
2211
2222222121
1112121111
nnnnnnnn
nn
nn
bababa
bababa
ba
baba
BA
CMA
BA
n
bababa
inin
n
i
ii
n
i
ii
n
i
)
2
(......)
2
()
2
(
=
1=
22
1=
11
1=
+
++
+
+
+
n
bababa
in
n
i
in
n
i
i
n
i
i
n
i
i
n
i
i
n
i
2
)(....)()(
1=1=
2
1=
2
1=
1
1=
1
1=
++++++
n
ba
ij
n
i
n
j
ij
n
i
n
j
2
)()(
=
1=
1=
1=
1=
+
2
)(
1
)(
1
=
1=
1=
1=
1= ij
n
i
n
j
ij
n
i
n
j
b
n
a
n
+
2
=
CMACMA
BA +
CMACMA
BA
=
So,
.
CMACMACMA
BABA
(iv)
Row and Column-Max-Average Norm and Max-Min Norm of Fuzzy Matrices
35
},{.....},{},{
},{.....},{},{
},{.....},{},{
=
2211
2222222121
1112121111
nnnnnnnn
nn
nn
bababa
bababa
bababa
BA
Now
ijijij
aba },{
and
ij
b
for all
j
i
,
.
i
n
i
ijij
n
i
aba
1
1=1=
)},({
and
ij
n
i
b
1=
for all
.
j
)(
1
)}],({[
1
1=
1=
1=
1= ij
n
i
n
j
ijij
n
i
n
j
a
n
ba
n
and
)(
1
1=
1= ij
n
i
n
j
a
n
CMACMACMA
BABA
Hence all the conditions of norm are satisfied by Column-max-average norm.
Note 1. Relation between Row-max-average norm and Column-max-average norm
is
CMA
T
RMA
AA =
.
Note 2. If
A
is symmetric i.e
T
=
then
CMARMA
AA =
.
7. Pseudo norm on fuzzy matrix
Pseudo norm on fuzzy matrices is a one type of norm but there is a difference between
norm on fuzzy matrix and pseudo norm on fuzzy matrix. Pseudo norm of a fuzzy matrix
fulfill the following conditions
1.
0A
for any fuzzy matrix
A
.
2. if
0=A
then
0=A
.
3.
AkkA
|=|
for any scaler
[0,1]
k
.
4.
BABA ++
for any two fuzzy matrix
A
and
B
.
5.
BAAB
for any two fuzzy matrix
A
and
B
.
Clearly except condition-2 all the condition of norm on fuzzy matrix and pseudo norm on
fuzzy matrix are same.
8. Max-min Norm
Max-min norm is an example of pseudo norm on fuzzy matrix. Here, first we will find
the maximum element in each row and then minimum of the maximum elements. In this
norm, we will use the new type of addition and multiplication of fuzzy matrices which
already we use in case of Row-Max-Average norm. 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
Lemma 3. All the conditions of pseudo norm of fuzzy matrix are satisfied b
Suman Maity
36
)(=
1=1= ij
n
j
n
i
MM
aA
.
Proof: Let us consider
nnnn
n
n
aaa
aaa
aaa
A
.....
.....
.....
=
21
22221
11211
and
nnnn
n
n
bbb
bbb
bbb
B
.....
.....
.....
=
21
22221
11211
)(=
1=1= ij
n
j
n
i
MM
aA
and
)(=
1=1= ij
n
j
n
i
MM
bB
(i) Clearly
0
MM
A
and if
0=A
then
0=
MM
A
.
(ii) According to the definition of max-min norm if
MM
A|>|
α
then
MMMMMM
AAA |=|=
αα
and if
MM
A|<|
α
then
.||=|=|
MMMM
AA
ααα
Therefore
MMMM
AA |=|
αα
for all
[0,1].
α
(iii) Now
+++
+++
+
++
2
.....
22
2
.....
22
2
.....
22
=
2211
2222222121
1112121111
nnnnnnnn
nn
nn
bababa
bababa
ba
baba
BA
)
2
(=
1=1=
ijij
n
j
n
i
MM
ba
BA +
}
22
{<
1=1=1=
ij
n
j
ij
n
j
n
i
ba
+
)}()({
2
1
=
1=1=1=1= ij
n
j
n
i
ij
n
j
n
i
ba
+
=
][
2
1
MMMM
BA +
=
MMMM
BA
Therefore
.
MMMMMM
BABA
(iv) Now
},{.....},{},{
},{.....},{},{
},{.....},{},{
=
2211
2222222121
1112121111
nnnnnnnn
nn
nn
bababa
bababa
bababa
BA
Row and Column-Max-Average Norm and Max-Min Norm of Fuzzy Matrices
37
If we denote
},{
ijij
ba
as
ijij
ba
then
).(=
1=1= ijij
n
j
n
i
MM
baBA
Now
ijijij
aba
and
ij
b
for all i,j.
ij
n
j
ijij
n
j
aba
1=1=
and
ij
n
j
b
1=
for all i.
)()(
1=1=1=1= ij
n
j
n
i
ijij
n
j
n
i
aba
and
)(
1=1= ij
n
j
n
i
b
MMMMMM
BABA
9. Conclusion
In this paper, we define two new types of operators on fuzzy matrices. Using this
operators we define different types of norm such as row-max-average norm, column-
max-average norm. Using these norm we can define conditional number to check
whether a system of linear equation is ill posed or well posed. Norm of fuzzy matrices
can take a effective contribution to solve a fuzzy system of linear equation.
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