Pacific Science Review B: Humanities and Social Sciences xxx (2016) 1—9

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Pacific Science Review B: Humanities and Social Sciences

journal homepage: www.journals.elsevier.com/pacific-science-review-b-humanities-and-social-sciences/

A study on pentagonal fuzzy number and its corresponding matrices

Apurba Panda*, Madhumangal Pal

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, 721102, India

ARTICLE INFO

ABSTRACT

Article history: Received 8 March 2016 Received in revised form 13 August 2016 Accepted 17 August 2016 Available online xxx

Keywords:

Pentagonal fuzzy number Arithmetic of pentagonal fuzzy number Pentagonal fuzzy matrices Pentagonal fuzzy determinant Nilpotent pentagonal fuzzy matrices

In this article, the notion of pentagonal fuzzy number (PFN) is introduced in a generalized way. A few articles have been published based on this topic, but they have some ambiguities in defining this type of fuzzy number. Here, we proposed the logical definition in developing a pentagonal fuzzy number, along with its arithmetic operations. Based on PFN, the structure of pentagonal fuzzy matrices (PFMs) is studied, together with their basic properties. Some special type of PFMs and their algebraic natures (trace of PFM, adjoint of PFM, determinant of PFM, etc.) are discussed in this article. Finally, the notion of nilpotent PFM, comparable PFM, and constant PFMs, with their many properties, are highlighted in this article.

Copyright © 2016, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University. Production and hosting by Elsevier B.V. This is an open access article under the CC

BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Decision making problems in the real world are very often uncertain or vague in most cases. Fuzzy numbers are used in various fields, namely, fuzzy process modelling, control theory, decision making, expert system reasoning and so forth. Previous authors' studies on fuzzy numbers highlighted the arithmetic and algebraic structure based on triangular fuzzy numbers and trapezoidal fuzzy numbers. Fuzzy systems, including fuzzy set theory (Zadeh, 1965) and fuzzy logic, have a variety of successful applications. Fuzzy set theoretic approaches have been applied to various areas, from fuzzy topological spaces to medicine and so on. However, it is easy to handle the matrix formulation to study the various mathematical models. Due to the presence of uncertainty in many mathematical formulations in different branches of science and technology, we introduced the concept of pentagonal fuzzy number (PFN) and corresponding pentagonal fuzzy matrices (PFMs). Several authors have presented results of the properties of a determinant, adjoint of fuzzy matrices, and convergence of the power sequence of fuzzy matrices. A brief review on fuzzy matrices is given below.

The concept of fuzzy matrices was introduced for the first time by Thomason (Thomason, 1977) in the article entitled convergence of power of fuzzy matrix; later, Hashimoto (Hashimoto, 1983a)

* Corresponding author. E-mail addresses: apurbavu@gmail.com, mmpalvu@gmail.com (A. Panda). Peer review under responsibility of Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University.

studied the fuzzy transitive matrix. The theoretical development of the fuzzy matrix was influenced through an article on some properties of the determinant and adjoint of a square fuzzy matrix proposed by Ragab et al. (Ragab and Eman, 1994). Moreover, some important results of the determinant of a fuzzy matrix were proposed by Kim (Kim et al., 1989). Several authors studied the canonical form and generalized fuzzy matrix (Hashimoto, 1983b; Kim and Roush, 1980), application of fuzzy matrices in a system of linear fuzzy equations (Buckley, 1991, 2001), etc. Some of the interesting arithmetic works on fuzzy numbers can be found in (Bhowmik et al., 2008; Dubois and Prade, 1979; Dubois and Prade, 1980). Conversely, some other articles studied different types of fuzzy numbers, namely, L-R type fuzzy number, triangular fuzzy number, and trapezoidal fuzzy number (Bansal, 2010). Thereafter, these types of fuzzy numbers were applied as a mathematical tool in the various fields of applied mathematics. The notion of a triangular fuzzy matrix was proposed for the first time by Shyamal and Pal (Shayamal and Pal, 2007) and was made familiar through introducing some new operators on triangular fuzzy matrices (Shayamal and Pal, 2004). The progression of fuzzy numbers became so fruitful that it spread into intuitionistic fuzzy matrices (Adak et al., 2012a; Adak et al., 2012b; Bhowmik and Pal, 2012; Bhowmik and Pal, 2008; Mondal and Pal, 2014; Pal, 2001; Pradhan and Pal, 2014a; Pradhan and Pal, 2014b; Pradhan and Pal, 2012; Shayamal and Pal, 2002) and interval valued fuzzy set theory (Mondal and Pal, 2015; Pal and Khan, 2005; Shayamal and Pal, 2006).

In this article, we introduce the notion of pentagonal fuzzy number in a well-defined manner by generalizing some other types

http://dx.doi.org/10.1016/j.psrb.2016.08.001

2405-8831/Copyright © 2016, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

A. Panda, M. Pal / Pacific Science Review B: Humanities and Social Sciences xxx (2016) 1—9

of fuzzy numbers and studied the basic arithmetic and algebraic properties of the pentagonal fuzzy number. In Section 2, several preliminaries regarding the fuzzy number are presented. In Section 3, fundamentals of the pentagonal fuzzy number are established. Based on the pentagonal fuzzy number, the concept of pentagonal fuzzy matrix (PFM) is presented in Section 4. Some works related to nilpotent PFMs, comparable PFMs, and constant PFMs are studied in the remaining sections.

2. Preliminaries

We first recapitulate some underlying definitions and basic results of fuzzy numbers.

Definition 1. Fuzzy set. A fuzzy set is characterized by its membership function, taking values from the domain, space or universe of discourse mapped into the unit interval [0,1]. A fuzzy set A in the universal setX is defined as A = (x,m(x);xeX). Here, mA:A/[0,1] is the grade of the membership function and mA(x)is the grade value ofxeX in the fuzzy set A.

Definition 2. Normal fuzzy set. A fuzzy set A is called normal if there exists an element xeX whose membership value is one, i.e., mA(x) = 1.

Definition 3. Fuzzy number. A fuzzy number A is a subset of real line R, with the membership function mA satisfying the following properties:

(i) mA(x) is piecewise continuous in its domain.

(ii) A is normal, i.e., there is a x0eA such that mA(x0) = 1.

(iii) A is convex, i.e., ma(1xi + (1-1)x2) > min(mA(xi),MA(x2)). V xi,x2 in X.

Due to wide applications of the fuzzy number, two types of fuzzy number, namely, triangular fuzzy number and trapezoidal fuzzy number, are introduced in the field of fuzzy algebra.

Definition 4. Triangular fuzzy number. A fuzzy number A = (a,b,c) is said to be a triangular fuzzy number if it has the following membership function

Thus, the triplet (a,b,c) forms a triangular fuzzy number under this membership function. Graphically, its membership function looks like a triangle, which is depicted in Fig. 1.

Definition 5. Trapezoidal fuzzy number. A fuzzy number A = (a,b,c,d) is called a trapezoidal fuzzy number if it possesses the following membership function

Graphically, the trapezoidal fuzzy number has a trapezoidal shape with four vertices (a,b,c,d), as depicted in Fig. 2.

However, real-life problems are sometimes concerned with more than four parameters. To resolve those problems, we propose another concept of the fuzzy number, called pentagonal fuzzy number (PFN). We discuss PFN in the next section.

Fig. 2. Trapezoidal fuzzy number.

3. Pentagonal fuzzy number

Due to error in measuring technique, instrumental faultiness, etc., some data in our observation cannot be precisely or accurately determined. Let us consider that we measure the weather temperature and humidity simultaneously. The temperature is approximately 35°C with normal humidity, i.e., the temperature is not perfect either more or less than 35°C, which affects normal humidity in the atmosphere. Thus, variation in temperature also affects the percentage of humidity. This phenomenon happens in general. This concept of variation leads to a new type of fuzzy number called the pentagonal fuzzy number (PFN). Generally, a pentagonal fuzzy number is a 5-tuple subset of a real number R having five parameters.

A pentagonal fuzzy number A is denoted as A = (a1,a2,a3,a4,a5), where a3 is the middle point and (a1, a2) and (a4, a5) are the left and right side points of a3, respectively. Now, we construct the mathematical definition of a pentagonal fuzzy number.

Definition 6. Pentagonal fuzzy number. A fuzzy number A = (a1,a2,a3,a4,a5) is called a pentagonal fuzzy number when the membership function has the form

where the middle point a3 has the grade of membership 1 and w1,w2 are the respective grades of points a2,a4. Note that every PFN is associated with two weights w1 and w2. To avoid confusion, we use the notation wiA for i = 1,2 to represent w1 and w2 as the weights of the PFN A.

3.1. Geometrical representation

From Fig. 3, it is clear that mA(x) has a piecewise continuous graph consisting of five points in its domain, forming a pentagonal shape. As chosen, the points in the domain have the ordering a1 < a2 < a3 < a4 < a5; a1,a2,a3,a4,a5eR. We have to choose the value of the membership function at a2,a4 in such a way that w1 > a^ I a1 and w2 > a^-a5.. Otherwise, the convexity properties of the fuzzy number fail for the pentagonal fuzzy number.

Remark 1. We define a pentagonal fuzzy number in a generalized way so thatwe can easily visualize two special fuzzy numbers, namely, triangular fuzzy number and trapezoidal fuzzy number, as follows:

Case I When w1 = w2 = 0, then the pentagonal fuzzy number is reduced to a triangular fuzzy number, i.e., A = (a1, a2, a3, a4, a5)y (a2,a3,a4); in this case

fl2 ~ X

Ü2 - a3;

mA{x) = <

a4 - x a4 - as '

x < a2 a2 < x < a3 x = a3 a3 < x < a4 x > a4

Fig. 1. Triangular fuzzy number.

A. Panda, M. Pal / Pacific Science Review B: Humanities and Social Sciences xxx (2016) 1—9

k < 0,kA = (ka5, ka4, ka3,ka2, kai)

(4) Multiplication: Let A = (ai,a2,a3,a4,a5) and B = (b-i,b2,b3,b4,b5) be two PFNs; then,

AB = (a1b1, a2&2, a3 &3, a4&4 , a5&5) with wi(AB) > max(wiA,wîb), i = 1,2.

(5) Inverse: We define the inverse of a PFN when all its components are non-zero. Suppose A = (ai,a2,a3,a4,a5) is a PFN; then,

i i i i i

If one of the components of a PFN becomes zero, then we cannot find its inverse.

(6) Division: The division of two PFNs A = (a1,a2,a3,a4,a5) and B = (b1,b2,b3,b4,b5) is approximated as the multiplication with

Fig. 3. Pentagonal fuzzy number.

Case II When w1 = w2 = 1, then the pentagonal fuzzy number becomes a trapezoidal fuzzy number, i.e., A = (a1 , a2, a3 , a4, a5)y (a1,a2,a4,a5),

0 for x < ai

x - ai ,

-— for ai < x < a2

a2 - a4 i - 2

1 for a2 < x < a4 a4 - x

and then mA(x)

for a4 < x < a5 a5 - a4 4 - 5

0 for x > x5

3.2. Arithmetic operations of PFN

Formation of an arithmetic operation is crucial in the study of fuzzy numbers; the author tries to establish some basic arithmetic operations of PFN. Note that every PFN is associated with two weights: w1 and w2. To avoid confusion, we use the notation wiA for i = 1,2 to represent w1 and w2 as the weights of the PFN A.

(1) Addition: Let A = (a1,a2,a3,a4,a5) and B = (b1,b2,b3,b4,b5) be two PFNs; then,

A + B = (a1 + &1, a2 + b2, a3 + 63, a4 + &4 , a5 + &5), with wi(A+B) > max(wiA ,wiB) for i = 1,2.

(2) Subtraction: We define the subtraction of two PFNs A = (a1,a2,a3,a4,a5) and B = (b1,b2,b3,b4,b5) as

A - B =(a1 - b1, a2 - b2, a3 - b3, a4 - b4 , a5 - b5), with wi(A-B) > max(wiA ,wiB) for i = 1,2.

(3) Scalar Multiplication: Let A = (a1,a2,a3,a4,a5) be a PFN and

keR be any scalar. If k > 0,kA = (ka1,ka2,ka3,ka4,ka5)

AzAB-i z(Ol Ol Ol 0±

B AB {fas' b4' fas ' b2 bj'

Note that a PFN A is divisible by B only when B is a non-null PFN having non-zero components.

(7) Exponent: The exponent of a PFN A = (ai,a2,a3,a4,a5) is defined as the power of its components. Anz(a1; a2; a3; al; aS), with n being a real number.

Remark 2. We choose the "max" relation of the weights of PFNs in all the above arithmetic operations because otherwise addition, subtraction, multiplication, division, etc., between two PFNs cannot be closed under these operations, i.e., the operations between two PFNs never produce another PFN. To verify, we assume that A = (-1,0,1,2,5) and B = (1,2,4,5,6), with w1A > 0.5, w2A > 0.7 and w1B > 0.3, w2B > 0.5; then, the value of w1(A+B), w1(A-B), w1(AB) are all greater than min(0.5,0.3) = 0.3, which violates the convexity condition of the pentagonal fuzzy number.

Definition 7. Positive PFN. A PFN A = (a1,a2,a3,a4,a5) is said to be positive if all its entries are positive. Similarly, A = (a1,a2,a3,a4,a5) is negative if all of its entries are negative.

Definition 8. Null PFN. A PFN A is called a Null PFN if all of its entries are zero, i.e., A = (0,0,0,0,0).

Definition 9. Null equivalent PFN. A PFN A = (a1,a2,a3,a4,a5) is said to be null equivalent if its middle entry is at the point 0, i.e., of the form (51,e1,0,e2,52), where 51-eiS0, 52-e2s0. It is denoted by 0.

Definition 10. Unit equivalent PFN. A PFN A is said to be a unit equivalent PFN when its middle entry is at 1, i.e., of the form (51,e1,1,e2,S2), where 51-e1s0, 52-e2s0.

From our previous arithmetic operations of PFN, we observed that subtraction of two PFNs with a common middle entry produces a null equivalent PFN, while their division yields another unit equivalent PFN. Additionally, we have the basic operations, i.e.,

a5 a4 a3 a2 ai

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addition and multiplication of PFNs are both commutative and associative, while multiplication is also distributive over addition.

Now, we construct a pentagonal fuzzy matrix whose elements are considered as pentagonal fuzzy numbers. This type of fuzzy matrix plays a vital role in fuzzy algebra.

4. Pentagonal fuzzy matrix

Definition 11. Pentagonal fuzzy matrix. A fuzzy matrix A = (aij)mxn of order m x n is called a pentagonal fuzzy matrix if the elements of the matrix are pentagonal fuzzy numbers, i.e., of the form

(anj,a2ij,a3ij,a4ij,a5ij).

Through classical matrix algebra, we achieve some algebraic operations of PFM. Let A = (aij) and B = (bij) be two PFMs of the same order; then, we have the following results:

(i) A + B = (aij + bj)

(ii) A-B = (aij-bij)

(iii) for A = (aij)mxr and B = (bij)rxn, we have AB = (cij)mxn, where (j = E n=1aikbkj for i = 1,2,...,m, j = 1,2,...,n.

(iv) AT = (aji), the transpose of A.

(v) kA = (kaij), where k is any scalar.

Some special types of pentagonal fuzzy matrices corresponding to classical matrices are now introduced in this section. However, in fuzzy matrix algebra, we define some other types of pentagonal fuzzy matrices and their algebraic properties.

Definition 12. Pure null PFM. A PFM is said to be a pure null PFM if all its entries are null PFNs, i.e., all the elements are (0,0,0,0,0). It is denoted by O.

Definition 13. Null equivalent PFM. A PFM A = (aij) is said to be a null equivalent PFM if all its elements are of the form aij = (51,e1,0,e2,52), where 51-e1s0, 52-£2s0. It is denoted as O.

Definition 14. Pure unit PFM. A square PFM A = (aij) is said to be a

pure unit PFM if aii = (0,0,1,0,0) and aij = 0, isjforallij = 1,2.....n. It is

denoted by I.

Definition 15. Unit equivalent PFM. A square PFM A = (aij) is said to be a unit equivalent PFM if aii = (51,e1,1,e2,52) and ay = 0, is j, where 51 • e1 s 0, 52 • e2 s 0 for all ij = 1,2.....n.

Definition 16. Pure triangular PFM. A square PFM A = (aj is

called a pure triangular PFM if either aij = 0 fori > j or aij = 0 fori<j. V i,j = 1,2,...,n.

When aij = 0 for i > j, then it is said to be a pure upper triangular PFM. Otherwise, for aij = 0, i < j, it is called a pure lower triangular PFM, i,j = 1,2,...,n.

Definition 17. Fuzzy triangular PFM. A square PFM A = aij is called a fuzzy triangular PFM if either aij = (S1, e1, 0, e2, 52) for i > j or

aij = (<51, £1, 0, £2, ) for i <j, where ^ • £1 s 0, ^ • £2 s 0.

Definition 18. Strictly fuzzy triangular PFM. A square PFM A = (aij) is called a strictly fuzzy triangular PFM if either aij = 0 for i > j or aij = 0 for i < j, 0 being the null equivalent PFN.

Definition 19. Symmetric PFM. A square PFM A = (aj is called a symmetric PFM if A = AT, i.e., aij = aji.

Definition 20. Pure skew symmetric PFM. A square PFM A = (aij) is called a pure skew symmetric PFM if A = -AT.

Definition 21. Fuzzy skew symmetric PFM. A square PFM A = (aij) is called a fuzzy skew symmetric PFM if A = -ATand aii = (51,£1,0,£2,52)

for all i = 1,2.....n, i.e., aij = -aji and att = (<51,£1,0,£2,<2),

51$£1s0, 52$£2s0. V i,j = 1,2,...,n.

5. Fundamental properties of PFM

Here, we introduce some fundamental properties of pentagonal fuzzy matrices. Here, we furnish the commutative and associative laws, which are well defined, for PFM under the arithmetic operations addition and multiplication.

Property 1. For any three square PFMs P,Q,R of the same order s x n, we have the following results:

(i) P + Q = Q + P.

(ii) P + (Q + R) = (P + Q) + R.

(iii) P + P = 2P.

(iv) P - P = O, a null equivalent PFM.

(v) P + O = P-O = P.

Property 2. Let P and Qbe any two PFNs of the same order and s,t be any two scalars. Then,

(i) s(tP) = (st)P.

(ii) s(P + Q) = sP + sQ.

(iii) (s + t)P = sP + tP, Vs, ts0.

(iv) s(P-Q) = sP-sQ.

Property 3. Let P and Q be any two PFMs such that P + Q and P- Q are well defined. Then,

(i) (PT)T = P.

(ii) (P + Q)T = PT + QT.

(iii) (P Q)T = OT-Pl

Property 4. Let P and Qbe any two PFNs of the same order and s,t be any two scalars. Then,

(i) (sP)T = sPT.

(ii) (sP + tQ)T = sPT + tQT.

Property 5. Let, P be any square PFM. Then,

(i) PPT and PTP are both symmetric.

(ii) P + PT is a fuzzy symmetric PFM.

(iii) P-PT is a fuzzy skew-symmetric PFM.

6. Trace of a PFM

Definition 22. Trace of a PFM. The trace of a square PFM A = (aj is defined as the sum of the elements of the principle diagonal. It is denoted by tr(A), i.e., tr(A) = £n=1aii.

Property 6. Let P = (pij) and Q = (qij) be two square PFMs of the same order m; then, the following holds well.

(i) tr(P + Q) = tr(P) + tr(Q).

(ii) tr(P) = tr(PT).

(iii) tr(P Q) = tr(Q P).

Proof. (i) Let P and Q be two PFMs of order m, where

pij = (p1 ij,p2ij,p3ij,p4ij,p5ij) and qij = (qnj,q2ij,q3ij,q4ij,q5ij). Now, tr(P) =

A. Panda, M. Pal / Pacific Science Review B: Humanities and Social Sciences xxx (2016) 1—9

E™lPii = Ya=i(Piü ,P2ii ,P3ii ,P4ii ,P5ii ) Ei=1 (q1ii,q2ii,q3ii, q4ii, q5ii).

tr(Q )=e m i Qu =

(iii) If P and Qare both pure upper triangular PFMs, then PQand QP both hold well.

Thus , tr(P + Q) = £(Pii + Qii) = E (Pii) + E (Qii)

= ^2(p1ii ,p2ii ,p3ii ,p4ii ,p5ii) i=1

+ ^2(q1ii, q2ii, q3ii, q4ii, q5ii) i=1

= tr(P) + tr(Q)

Hence the result.

(ii) We know that the principle diagonal of a PFM remains invariant under transposition. Hence, the proof is obvious.

(iii) We know that for any two PFM of the same order, their multiplication is well defined. Let P = (pij) and Q = (qij) be two PFMs of the same order m, where (pj = (p1ij,p2ij,p3ij,p4ij,p5ij) and (qij) = (q1ij,q2ij,q3ij,q4ij,q5ij). Again, let C = (j where (cj = T,m=1pirqrj, for ij = 1,2,...,m. Now, tr(C) = Y,m=1cii = £|=1 (En=1pirqri). Again, let D = (d,j) = QP, where dj = £ ^=1 qirprj, for ij = 1,2,...,m. Therefore, tr(D) = tr(Q■P) = Ei=1dii

n f m \

= ^[j2qirPri 1

= E E Pirqri (interchanging the dummy indices i and r)

r=A i=i J

= tr(PQ) = tr(C)

Hence the proof.

Property 7. The product of two pure upper triangular PFMs oforder k x k is a pure upper triangular PFM.

Proof. Let P = (pij) and Q = (qj be two pure upper triangular PFMs of the same order k, where (pj = (p1ij,p2ij,p3ij,p4ij,p5ij) and (qij) = (qnj-,q2ij,q3ij,q4ij,q5ij). Because P,Q are both upper triangular

PFMs, pij = (0,0,0,0,0) and qij = (0,0,0,0,0) for i > j, ij = 1,2.....k. Let

N = P Q = (n,j); then, (pij) = £kr=1ptqr j = £k=1(pnj,p2ij,p3ij,p4ij, p5ij)(q1ij,q2ij,q3ij,q4ij,q5ij). Now, it is enough to establish that

(nij) = (0,0,0,0,0) for i > j, ij = 1,2.....k. For i > r, we have

pir = (0,0,0,0,0), r = 1,2,...,i-1, and qir = (0,0,0,0,0), r = i,i + 1,...,k

Therefore , (nj =J2 PirQrj = PirQrj + Y1 PirQrj

r=1 r=1

(0 ,0,0 ,0 ,0).

Now , nii = pirqri

i-1 ^ = pirqrj + pirqrj r=1 r=i-1

= pu-qu = (0 ,0 ,0 ,0 ,0)

because pir = (0,0,0,0,0), r = 1,2.....i-1 and qir = (0,0,0,0,0),

r = i,i + 1,...,k. Hence the result.

Property 8. The product of two pure lower triangular PFMs is also a pure lower triangular PFM.

Property 9. Let P be any square PFM of order m.

(i) If P is a pure upper triangular PFM, then PT is a pure lower triangular PFM.

(ii) If P is a pure lower triangular PFM, then PT is a pure upper triangular PFM.

7. Determinant of a PFM

In this section, we introduce another important algebraic property, i.e., the determinant of a PFM, together with its several postulates. Also we mention the characteristic of an adjoint, cofactor, minor, etc., and classify their properties.

Definition 23. Determinant of a PFM. The pentagonal fuzzy determinant of a pentagonal fuzzy matrix A of order n x n is denoted

by det(A) or |A| and defined as |A| ^^ (sgns$ aisi), where

S2S„ i=1

aisi = (a1isi,a2isi,a3isi,a4isi,a5isi) are PFNs and Sn denotes the symmetric

group of all permutation of indices 1,2.....n. Additionally, sgn is the

signature of the permutation, defined as sgn s = 1 or -1 if the permutation is even or odd, respectively.

There are several products and additions of PFNs in the computation of det(A), and the value of PFNs generates another PFN. Thus, the determinant value of a PFM yields a pentagonal fuzzy number.

Definition 24. Minor. Let A be a square PFM of order n x n. The minor of an element aij in det(A) is the determinant of order (n-1) x (n-1), which can be obtained by deleting the ith row andjth column from A. The minor of A is denoted by Mij.

Definition 25. Cofactor. Let A be a square PFM of order n x n. The cofactor of an element aij in A is denoted by Aij and is defined by

Aj = (-1)i+jMij.

Definition 26. Adjoint of a PFM. Let A = (aij)nxn be a square PFM. The adjoint of a PFM A is denoted by adj(A) and is defined as bij = \Aji\, where \Aji\ is the determinant of a (n-1) x (n-1) PFM formed by deleting row j and column i from A, i.e., B = adj(A). This can be defined as

adj(A) = B :

reSn-n. ten ninj J

itp(t)

where nj = 1,2.....n-j and p is an arbitrary permutation chosen from

the set of all permutations Sni nj of set ni over nj.

Here, det(A) contains n! terms, out of which 2 are positive and the remaining same number of terms are negative. All these n! terms contain n quantities at a time in product form, subject to the condition that from the n quantities in the product, exactly one is taken from each row and exactly one is taken from each column. Another way of representing the pentagonal fuzzy determinant of a PFM A = (aij) is to expand it to the form £ aijAji, i = (1,2,.. .,n), where Aij is the cofactor of aij in det(A). Thus, the pentagonal fuzzy determinant is the sum of the product of the elements of any row (column) and the co-factors of the corresponding elements of the same row (column).

In classical matrix algebra, the value of a determinant can be computed using any one of the above-mentioned two processes, with both yielding the same result. The simplest way to determine the value of the pentagonal fuzzy determinant is given by the formula as in definition. We now study the important properties of pentagonal fuzzy matrices.

Property 10. Let P = (pij) be a PFM of order m x m.

(i) If all the elements of a row (column) of P are (0,0,0,0,0), then jPj = (0 ,0 ,0 ,0 ,0).

(ii) If a row (column) is multiplied by a scalar l, then det(P) is also multiplied by l.

6 A. Panda, M. Pal / Pacific Science Review B:

(iii) If P is a pure triangular PFM, then |P| = ]^[(p1ii,p2ii

, p3ii, p4ii, p5ii). i=1

Proof. (i)Let P = (pij)mxm be a square PFM, where (pij) = (p1ii,p2ii,p3ii,p4ii,p5ii). We define the determinant in the following way: Ei(A) = T,jL1aijAij = EjLl(a1j, a2ij, a3ij, a4ij, a5ij)Aij,

where Aij is the cofactor of pij in det(P). Obviously, E1(P) =E2(P) = ••• = Em(P) = jPj. Now, assume that all the elements of the rth row, 1 < r < m, are pure null PFN. Then, Er(P) = (0,0,0,0,0). Because prj = (0,0,0,0,0) for all j = 1,2,...,m,

jPj= Er(P) = (0,0,0,0,0). Hence the result.

(ii)If l = 0, then P has a zero row (column). Thus, |P| = (0,0,0,0,0). Thus, the result is obviously as follows.

Let Q = (qij) be a square PFM of order m obtained from P by multiplying its rth row by a non-zero scalar l. Then, clearly qij = (q1ij ,q2ij,q3ij,q4ij,q5ij) = 1(P1j,P2j,P3j,P4j,P5j) = (1P1j,1P2j,1P3j,1P4j,1P5j) when l is a positive scalar and qij = (1p5ij,1p4ij,1p3ij,1p2ij,1p1ij) when l is a negative scalar.

Now, \Q\ = J2 sgn^q1(1ff1),q2(1ff1),q3(1ff1)>q4(1ff1)>q5(1a1)) —

<reSm v y

(q1(rar)i q2(rar)j q3(rar), ^(nr), q5(rar)J —

{q1(nan), q2(mam)> q3(nan)j q4(nari), q5(mam)^

= J2 sgn^p1(1a1)> p2(1a1)> p3(1a1), p4(1a1), p5(1a1)) "" aeSm

(p1 (rar), 1p2(rar), 1p3(rar), 1p4(rar), 1p5(rar)^ ■ ■ ■

(p1 (mam) , p2(nan) , p3(mam) , p4(mam) , p5(mam) m

= l J2 sgna-JJ Piai aeSm i=1

= 1\P\

When l is a negative scalar,

\Q\ = E sgna( q1(1a1), q2(1a1), q3(1a1), q4(1a1^ q5(1a1)) —

aeSm 7

(1p5(rar)J 1p4(rar)J 1p3(rar)J 1p2(rar)J 1p1(rar)) ■■•

(p5(mam) (p4(mam), p3(mam), p2(mam), p1 (mam))

= J2 sgna(p1(1a1)J p2(1a1)J p3(1a1), p4(1a1^ p5(1a1)) — aeSm

(1p1 (rar), 1p2(rar)J 1p3(rar)J 1p4(rar)J 1p5(rar)) ■■•

(p1 (mam), p2(mam), p3(mam), p4(mam), p5(mam m

= l J2 sgna-JJ Piai aeSm i=1

Hence the result.

(iii)Let P = (aij)mxm be an upper (lower) triangular PFM. Then, for i < j, pij = (0,0,0,0,0). Now consider a term t in \P\; then,

t = n(P1 iai, P2 iai, P3 iai, P4 iai, P5 iai). Let s(1) s 1, i.e., 1 < s(1), so that

P11 s1 = 0, P21s1 = 0, P31s1 = 0, P41s1 = 0, P51s1 = 0. Consequently, pisi = 0 for i = 1. Again, let s(1) = 1 but s(2)s2; then, p2s2 = 0. Hence, t = (0,0,0,0,0). This means that for each term, ^ = (0,0,0,0,0), if s(1)s1, s(2)s2. Preceding in this way, we have for s(i)si t = (0,0,0,0,0). Therefore,

\P\ = II(P1ii, P2ii, P3ii, P4ii, P5ii). i=1

This implies that the product of the diagonal entries is the value of the determinant for a triangular PFM.

Humanities and Social Sciences xxx (2016) 1—9

Property 11. The determinant of a diagonal PFM is the product of its diagonal entries.

Property 12. If any two rows (columns) of a square PFM A are interchanged, then only the sign of determinant jAj of A changes.

Proof. Let A = (aij) be a square PFM of order n x n. If P = (pj is obtained from A by interchanging the rth and sth row (r < s) of A, then it is clear that pij = aij, isr, iss and pj = asj, psj = aj Now, ^ =

E sgna(p1a(1)P2a(2)—Pra(r)—Psa(s) —Pna(n))= E sgna(aM1)

aeSn aeSn

a2s(2) ■ ■ ■ars(s) ■■■ass(r) ■■■ans(n) )

1 2 1 2

S = sg =

Then, g is a transposition of interchanging r and s. Thus, g is an odd permutation; thus, sgnl = -1. Let gs = 5. As s runs through all permutations on (1,2,...,n), < also runs over the same permutations because s1g = S2g or s1 = S2.

1 2 — r — s — n a(1) a (2) — a(r) — a(s) — n 12 — r — s — n 12 — s — r — n s(r) = s(s), s(s) = s(r). Because g is an odd permutation, 5 is even or odd if s is even or odd, i.e., sgnd = -sgns.

Then, | P| = £ sgns

.Therefore, s(i) = i; isr,s;

Uaisi) i=1 )

Uaisi) i=1 )

= - E sgn5

= -\A \.

Hence the result.

Property 13. If A is a square PFM, then the determinant value of A equals to that of its transpose, i.e., \A \ = \AT \.

Proof. Let A = (aij) be a square PFM of order n and let P = AT be the transpose of A. Then, by the definition of a pentagonal fuzzy

determinant, we have \P\ = (sgna- pia(i)) = y] (sgna- ]~[

aeSn i=1 aeSn i=1

aa(i)i). Let f be a permutation on 1,2.....n such that fs = I, I being the

identity permutation. Thus, f = s-1. Let s(i) = j; then, i = s(j)-1 and

as(i)i = ajs(j), V ij.

Therefore; P = (sgna p1a(i)i; p2a(i)i; p3a(i)i; p4a(i)i; p5a(i)i

aeSn i=1 v

= E sgna n\a1ja(j), a2ja(j), a3ja(j), a4ja(j), a5ja(j) aeSn j=1 y

n ( ■

= E sgna n ia1ia(i), a2ia(i), a3ia(i), a4ia(i), a5ia(i) ) aeSn i=1 v

[interchanging indices] = \A \.

Hence the result. Property 14. For a square PFM A of order n:

(i) If A contains a zero row, then adj(A)A is a null equivalent PFM.

(ii) adj(AT) = [adj(A)]T.

Proof. (i) Let A = (aij) be a square PFM of order n x n, where aij = (a1ij,a2ij,a3ij,a4ij,a5ij). Let B = adj(A); then, by the definition of the adjoint of a PFM, the (ij)th element of bj of B is \Aj\, where Aj is the sub matrix obtained from A by suppressing the ith row and jth

A. Panda, M. Pal / Pacific Science Review B: Humanities and Social Sciences xxx (2016) 1—9

column, i.e., Aij is the cofactor of aij in A. Without loss of generality, we assume that the kth row of A is the zero row. Therefore, the elements of the kth row are of the form (eikj,5ikj,0,52kj,e2kj), 5uj- 52kjS0 and e1kj-e2kjS0 for all j. Then, all elements of adj(A) are of the form

|A,j| = (eikj, dikj, 0, &2kj, £2kj), except jsk. Let P = (adjA)A. Then, the (ij)th element of P is of the form Pij = Em=i\Aim\amj= E \Aim\amj + Y,m=i\Aik\akj. Now all \Aim\,

msk, are of the form of the null equivalent PFN. Hence, pij is of the

form of a null equivalent PFN for all ij = i,2.....n. Thus, (adjA)A is a

null equivalent PFM.

(ii) The proof for this part obviously follows from the definition.

Property 15. Let Abe a square PFM of order n.

(i) If A is a symmetric PFM, then adj(A) is a symmetric PFM.

(ii) If A is a null equivalent PFM, then adj(A) is also a null equivalent PFM.

(iii) If A is a pure unit PFM, then adj(A) is a unit equivalent PFM.

Proof. (i) From Property i5, it is clear that the adjoint property for a PFM preserves transposition, i.e.,adj(AT) = [adj(A)]T. Because A is a symmetric PFM, AT = A. Now [adj(A)]T = adj(AT) = adj(A). Hence, adj(A) is a symmetric PFM.

(ii) Let A be a null equivalent PFM of order n; then, all the elements of A are null equivalent PFNs, i.e., aij = 0. Again, adj(A) is the transpose of the cofactor matrix of A. Thus, adj(A) = [Aij]T = (-i)l+jMji. Additionally, My is the determinant of an (n-i) x (n-i) order matrix, deleting the ith row and jth column from A. Because each aij is a null equivalent PFN, the cofactors of the elements of A are null equivalent PFNs and hence its transpose. Finally, we conclude that adj(A) is a null equivalent PFM.

(iii) Because A is a pure unit PFM of order n, its diagonal entries are of the form aii = (0,0,i,0,0) and aij = 0, isj. It is now clear that the cofactors of diagonal elements are nothing but the determinant value of a pure unit PFM of order (n-i) x (n-i), which is a unit equivalent PFN and null equivalent PFN of non-diagonal elements. Hence, adj(A) is a unit equivalent PFM.

8. Fuzzy comparable PFM

In this section, we newly introduce fuzzy comparable PFM to resolve the fuzzy order preference problems. Between any two matrices, there is an ordering relation: either they are equal or different. This deals with pairwise comparison of matrices under elementary-order priority. Pairwise comparison is applied whenever the decision maker is not sure regarding the evaluation of relative importance.

Here, we adopt the concept of fuzzy order relations between two elementsofafuzzyset,i.e.,forafuzzysetTandx, y e T; theorderrelation denoted by order lattice " < " holds whenever x < y or y < x.

Definition 27. Fuzzy comparable PFM. Let P and Q be two PFMs of order n x n. We say that P is comparable to Qif either P < QorQ < P, i.e., when pij < qij0P < Q or pij > qij 0 Q < P. When both are equal, we called them equivalent PFM.

Property 16. Let P and Qbe two PFMs of order m x n. Then, we have the following:

(i) For any PFM T of order m x n, we have P < Q, which implies P + T < Q + T and vice versa.

(ii) For any PFM R of order n x p, we have P < Q, which implies P R < QR.

(iii) If Pi < P2 and Qj < Q2, then their product is also comparable, i.e., Pj • Qj < P2 • Q2 for the compatible matrix product of Pj • Qj and P2 Q2.

Proof. (i) Because P < Q, then we have Pj < qij, which implies

(Piij,P2ij,P3ij,P4ij,P5ij) < (qiij,q2ij,q3ij,q4ij,q5ij)

Now, P + T = Pij + tij = (pjij, P2ij, P3ij, P4ij; P5ij) + (tJij> t2ij; t3ij; t4ij; t5ij)

0Pij + tij < qij + tij [because P < Q] 0P + T < Q + T

Conversely, let P + T < Q + T. Then,

Pij + tj < qj + tj

0 (Plij> P2ij, P3ij, P4ij, P5ijJ + (tJij> t2ij, t3ij, t4ij, t5ij) < (qjij> q2ij; q3ij; q4ij; q5ij) + (tlij> t2ij; t3ij; t4ij; t5ij)

0P < Q.

(ii) Here, P, Q are comparable PFMs of order m x n. Let R be any PFM of order n x p.

Because Pij < qij, (Pik$rkj) < (q^rj V i,j,k. Therefore^JJ=j (Pik$rkj) <En=j(qik$rkj), i.e., P R < Q R. Hence the result.

(iii) By a similar approach, we can get the result. It is observed that (ii) is the particular case of (iii) only when Qj = Q2.

9. Some results of nilpotent PFM

Nilpotent matrices are of great importance in fuzzy algebra. Here, we define the nilpotent matrix in the fuzzy sense based on the pentagonal fuzzy matrix and study some properties that hold especially for pentagonal fuzzy matrices.

Definition 28. Nilpotent PFM. Let A = (üij)nxn be a square PFM of order n. A is said to be nilpotent for the index l if l is the least positive integer such that A1 = O.

Property 17. Let A and B be two nilpotent PFM of index m,n, respectively. Then, A B and A + B are both nilpotent whenever AB = BA.

Proof. Let A and B be two nilpotent PFMs of index m,n, respectively. Then, Am = O and Bn = O. Again, let k = lcm(m,n).

Now, {A-B)k = {A-B){A-B){A-B)---k times. = {AABB)[{A-B){A-B)/{k - 2) times.] = (A2-B2)[(A-B)(A-B)/(k - 2) times.] = (A3-B3)[(A-B)(A-B)---(k - 3) times.] = Ak$Bk , [because A$B = B$A] = O.

Thus, the product of two nilpotent PFMs is also a nilpotent PFM. Second part. The nilpotency of A + B can be shown directly from the Binomial theorem.

Property 18. Every strictly fuzzy triangular PFM of order n is nilpotent for index n.

Proof. Let us consider an n x n strictly fuzzy upper triangular

PFM A = (aij), where (a,j) = (ai[j,a2[j,a3lj,a4y,a5[j). Let A be of the form

( ~ Ö ■ • ■■• ö\

A = «21 Ö ■ ■ ö

Uni «n2 ■ • «nn-1 öj

Let the entries of A2 be ft/I then, (ßij) = Yl 1k=1aikakj, f°r

A. Panda, M. Pal / Pacific Science Review B: Humanities and Social Sciences xxx (2016) 1—9

ij = 1,2.....n. Because A is strictly upper triangular, aij = 0, i < j. Thus,

by looking at the entries of A, we have bj = 0, V i < j. Now, for

j = i-1, bii-1 =E^aik^n = a1ka1i-1 + ai2aki-1 + -ainani-1. Because each ain or ai n-1 will lie on or above the principle diagonal

/5 0 ...... 0\

0 0 ...... 0

b21 0 • 0

of A, bu-1 = o. A2 =

\bn1 ...... bnn-2 0 0/

That is, bij = 0, for i-1 < j. We see that each time we raise A to its power, the next diagonal under the principle diagonal becomes zeros. Again, let us assume that this occurs for the power of A, i.e., for Ak, the kth diagonals, including the main diagonal, become zeros. We assume the entries ofAk to be gj;then, gj = 0 for i-k +1 <j.Nowitis sufficient to prove that the (k + 1)th power of A is a strictly upper triangular PFM having the next k diagonals until the principle di-

agonal vanishes. Let the elements of Ak+1 be 5j because Ak+1 = AAk. Thus, we have 5ij = n=1 aikgkj because for all i < j, aij = 0. Thus, it is

clear that 5j = 0, V i < j for ij = 1,2.....n. Again, 5j = ln=1aikgkj.

We see that for i-k + 1 < j, 5ii-k+1 =

ang1i-k+1+ai2g2i-k+1 + •■■ + aingni-k+1 V i,j =l,2 —,n. From

the above expression, we have ain or gni-k+1 vanishes for gj = i-k + 1 < j.

Therefore, Ak+1 gives the PFM whose (k + 1)th diagonals under the main diagonal are zero, i.e., of the form

gk+11 V gn 1

gnn-k+1 5/

Hence, the result is true for n = k + 1 when it is true for n = k. Additionally, the result is true for n = 2. Therefore, by mathematical induction, we conclude that the result is true for all n. Thus, An finally produces a pure null PFM, i.e., An = O.

Hence, a strictly fuzzy triangular PFM of order n is nilpotent for the index that is exactly n.

Property 19. (i) If A and B are both strictly fuzzy triangular PFMs, 'A C

then the block matrix

(ii) Generally,

is nilpotent.

is nilpotent whenever Aii's

are strictly fuzzy triangular PFMs.

Proof. (i) Let us introduce the concept of block matrix P of the

form P =

A C O B

where A,B are strictly fuzzy triangular PFMs of

order m, n, respectively. Thus, based on the property, A and B are both nilpotent for index m and n, respectively.

Now; P2

A2 AC + BC

A C\fA C

O B) \O B J V 5 B2 A2 AC + BC\(A C 5 B2 JyO B A3 A2C + B2C + ABC 5 B2

Thus, note that when we raise the power to P, the elements p11 and p22 increase their power that of element p12, i.e., in general, we Ak

have Pk =

, with k being a positive integer and assuming

l = lcm(m,n) (say), (Pk)x =

which is a

"a" as the value of the element p21 in Pk. A and B are nilpotent for index m,n, respectively; therefore, Am = O, Bn = O. Taking

f AkX a

strictly fuzzy triangular PFM; hence, based on Property 18, P is a nilpotent PFM. Hence the proof.

Second Part. (ii) It follows from the previous properties.

10. Singular and constant PFM

Definition 29. Singular PFM. A square PFM A is said to be singular if the determinant value is a pure null PFN, i.e., \A\ = (0,0,0,0,0).

Definition 30. Semi-singular PFM. A square PFM is called semisingular when its determinant value produces a null equivalent PFN, i.e.,\A \ = (51, £1, 0, £2, 52), <1 -£1 s0, <2 -£2 s0. V i,j = 1,2, —, n.

Definition 31. Constant PFM. A square PFM A = (aj) of order n x n is called a constant PFM if all the rows are equal to each other, i.e.,

(a1ij, a2ij, a3ij, a4ij, a5ij) = (a1rj, a2rj, a3rj, a4rj, a5rj) V i, r, j.

Example 1. For example, we consider a constant square PFM A of order 3 as

((-1, 0,1, 2, 4) (0,1; 2,4, 5) (1; 2,3,4, 5)' A = (-1, 0,1, 2, 4) (0,1, 2,4, 5) (1, 2,3,4, 5) V(-1, 0,1, 2,4) (0,1, 2,4, 5) (1, 2,3,4, 5),

Property 20. Let A and B be two constant PFMs of the same order. Then, the following holds.

(i) A + B is a constant PFM.

(ii) A- B is also a constant PFM.

Proof. (i) Let A = (ay) and B = (bij), where (aij) = (a1ij,a2ij, a3j,a4j,a5j) and bj = (b1ij,b2ij,b3ij,b4ij,b5ij) are two constant PFMs of order n. Then, (a1ij,a2ij,a3ij,a4ij,a5ij) = (a1rj,a2rj,a3rj,a4rj,a5rj) and

(b1ij,b2ij,b3ij,b4ij,b5ij) = (b1rj,b2rj,b3rj,b4rj,b5rj)

Let C= icij) = iaj + bij) = (a1jJ a2ijJ a3jJ a4ijJ a5ij) +

(bw, b2ij, b3ij, b4ij, b5j). i, r,j = 1, 2, —, n = (a1rjJ a2rjJ a3rjJ a4rjJ a5rj) + (b1rjJ b2rjJ b3jJ b4rjJ b5rj)

[because A, B are constant.]

= {c1rjJ c2rjJ c3rjJ c4rjJ c5ij) = crj V h rJ j.

Thus, the rows of (A + B) are similar to each other. Hence the proof.

(ii) The proof for this part follows from the definition. Property 21. Let A be any square PFM; then, the following results hold:

(i) [adjA]T is constant.

(ii) A.adjA is constant.

(iii) A.adj(AT) is constant.

(iv) (AT-adjA)T is constant.

(v) (AT-adjA) is constant.

(vi) The determinant value of a constant PFM is a null equivalent PFM.

Proof. (i) Because A = (aj is a constant PFM of order n x n, where (aij) = (a1j,a2j,a3j,a4j,a5j), then (a1j,a2j,a3j,a4j,a5j) = (a1rj,a2rj,a3rj,a4rj,a5rj). Now, let bij = adj(A). Then,

A. Panda, M. Pal / Pacific Science Review B: Humanities and Social Sciences xxx (2016) 1—9 9

by = J2 n^m

P2Sn.n. t2Hj i j J

bik = J2 I! atp(t)

P2Sn¡nk t2nk

It is obvious that by = bik because the numbers p(t) of the columns cannot be changed in the two expansions of by and bik. Thus, [adjA]T is constant.

(ii) Because A is a constant PFM, both Ajk = Aik and det(Ajk) = det(Aik) hold for every i, je{1,2.....n}. Again, let C = A adjA. Thus,

cij = J2 aik-det(Ajk) = J2 aik$det(Aik) k=1 k=1

Additionally, from the definition of the determinant of a PFM in terms of the adjoint, det(A) = £n=1aik-det(Aik). Thus, cy = det(A). Thus, similar to the fuzzy matrix, this result also holds well for a pentagonal fuzzy matrix.

(iii)/(iv) These proofs can be obtained via transpose operations and also by transposing of the constant PFM to remain constant.

(v) We earlier proved (Property 14) that the determinant of a PFM A having a zero row is a null equivalent PFN. Additionally, for a constant PFM A, the rows are equal to each other. Thus, one can get a zero row via the elementary row operation. Hence the result.

11. Conclusion

In this article, special attention is paid to the pentagonal fuzzy number (PFN) and the corresponding pentagonal fuzzy matrix (PFM), along with the related mathematical expressions. Based on applying elementary algebraic operations to the PFM, we studied various types of PFM and their properties (determinant, adjoint, trace, etc.). Second, this paper addresses the nature of nilpotent PFM and comparable PFM, with some interesting properties. There are several opportunities to develop the applications of such pentagonal fuzzy number. We are trying to investigate such applications.

Acknowledgements

The author thanks the supervisor and DST INSPIRE for granting a fellowship (N0-lFi50087) during this research work.

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