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The determinant and adjoint of a square fuzzy matrix
Fuzzy Sets and Systems 61 (1994) 297-307 North-Holland
297
The determinant and adjoint of a square fuzzy matrix M.Z. Ragab and E.G. Emam Department of Mathematics, Zagazig University, Zagazig, Egypt Received March 1993 Revised July 1993
Abstract: The theory of the determinant and adjoint of a square fuzzy matrix will be studied. Also we define the circular fuzzy matrices and show that some properties of a square fuzzy matrix (such as reflexivity, transitivity, circularity) are carried over to the adjoint of the matrix. Finally we show how to construct a transitive fuzzy matrix from a given one through the adjoint matrix.
Keywords: Determinant of a square fuzzy matrix; adjoint of a square fuzzy matrix.
1. Introduction
There are some papers on the determinant theory of square fuzzy matrices (see Kim [2, 3]). In the present work we study and investigate some properties of the determinant of a square fuzzy matrix. However, the results are similar to the crisp case. The adjoint of a square fuzzy matrix is defined by Thomason [5] and Kim [3]. We state a formula for the adjoint matrix (see Remark 1) and use this formula anywhere in the paper. In this paper we define the circular fuzzy matrices and show that some properties of a square fuzzy matrix such as reflexivity, transitivity and circularity are carried over to the adjoint matrix and we establish some results including that A adj A/> JA] I, (adj A ) A >i IAI I and IAI = ladj AI where IAI denotes the determinant of A and adj A denotes the adjoint matrix of A. At the end of this paper we prove that A adj A is transitive for any square fuzzy matrix A. However this result enables us to construct a transitive fuzzy matrix from a given one through the adjoint matrix. 2. Preliminaries
We give here some definitions which are applied in the paper. Note that the elements of a fuzzy matrix takes its values from the unit interval [0, 1]. Definition 2.1 [4,5]. For fuzzy matrices A = [aij] (n x n ) , B = [bij] (n × p ) and C = [qj] (n × p ) , the following operations are defined
B + C = [bij + cij] where bit + cij = max(b/h qj), A B = I ~ jl aikbkJ =
where aikbkj = min(a~, bkJ) ,
A ' - - [aj~] (the transpose of A),
A ~ = la~v~'], A k-~' = A k A (k = O, 1, 2 . . . . ), A ° = I,, where In is the usual identity matrix, B<~C i f f b ~ j ~ < q j f o r a l l i ~ { 1 , 2 . . . . . n } a n d j ~ { 1 , 2 , . . . , p } . Correspondence to: Dr. M.Z. Ragab, Department of Mathematics, Zagazig University, Zagazig, Egypt. 0165-0114/94/$07.00 © 1994~Elsevier Science B.V. All rights reserved SSDI: 0165-0114(93)E0188-X
M.Z. Ragab, E.G. Emam
298
/ The determinant and adjoint of a square fuzzy
matrix
Definition 2.2 [4]. L e t A be an n / n fuzzy matrix. T h e n A is said to be s y m m e t r i c iff A = A ' , reflexive iff A / > In, transitive iff A 2 ~< A, similarity iff it is reflexive, s y m m e t r i c and transitive, and i d e m p o t e n t iff AZ=A.
Definition 2.3. An m × n fuzzy matrix A is said to be constant if aik equal to each other.
=
ajk for all
i, j, k,, i.e., its rows are
3. The determinant of a square fuzzy matrix
Definition 3.1 [3]. T h e d e t e r m i n a n t IAI of an n z n fuzzy matrix A is defined as follows: IAI = ~
al~(1)az,~(2)"'an,~(n)
o'ESn
w h e r e Sn d e n o t e s the s y m m e t r i c g r o u p of all p e r m u t a t i o n s of the indices (1, 2 , . . . , n).
Proposition 3.2. I f a f u z z y matrix B is obtained f r o m an n × n f u z z y matrix A by multiplying the i-th row o f A (i-th c o l u m n ) by k ~ (0, 1], then k IAI = Inl.
Proof. By definition IBI-- ~
blo-(1)b2o-(2)"
""
bn,,(,,)
o-~Sn
=
~
alo-(1)a2tr(2)
• • •
kaio-(i) • • • am~(n )
~,ES n
=
k
Z
al~(1)a2~(2)
" " "
a i d ( i ) " " " anon(n)
o- ~ S n
= k IAI
Proposition 3.3. Let A be an n x n f u z z y matrix. Then we have: (1) if A contains a zero row (column), then IAI = 0, n a (2) if A is triangular, then [AI = I-I~'=l aii where ]-Ik=l k = ala2"
• an.
Proof. (1) E a c h t e r m in IAI contains a factor of each row ( c o l u m n ) and hence contains a factor of zero row (column), so that each t e r m of IAI is equal to zero and c o n s e q u e n t l y IAI = 0. (2) S u p p o s e A = [au] is triangular f r o m below, i.e., a u = 0 for i < j . T a k e a t e r m t of IAI t =
alo-(1)a2o-(2
) • • •
ang(n).
Let o.(1) ~ 1, so that 1 < o.(1) and so a1,,(1) = 0 and t = 0. This m e a n s that each t e r m is zero if o.(1) ~ 1. Let n o w o-(1) = 1 but o'(2) ~ 2. T h e n 2 < o'(2) and a2~(2) = 0 and t = 0. This m e a n s that each t e r m is zero
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
299
if 0-(1) ¢ 1 or 0-(2) ~ 2. However, in a similar manner we can see that each term for which 0-(1) ¢ 1 or 0-(2) ~ 2 . . . or 0-(n)¢ n must be zero. Consequently
qAI
=
a l i a 2 2
"
"
"
ann = ~ I aii" i=l
From the above proposition it is clear that III= 1.
Theorem 3.4 [2]. L e t A and B be two f u z z y matrices. Then IABI >1 IZl IBI.
Example 1. For a fuzzy matrix
A=
0.5 0.6
0.3 0.2
0.8] 0.9
0.0
0.7
0.4
we calculate the determinant IAI as follows: 001~ JA[ =0.5
0.9 + 0.3 0.6 0.4 0.0
0.9 0.6 0.4 +0"8 0.0
0.2 0.7
= 0.5(0.2 + 0.7) + 0.3(0.4 + 0.0) + 0.8(0.6 + 0.0) = 0.5(0.7) + 0.3(0.4) + 0.8(0.6) = 0.5 + 0.3 + 0.6 = 0.6.
4. The adjoint of a square fuzzy matrix This is the principal section in this paper. We begin with the following.
Definition 4.1 [3]. The adjoint matrix of an n x n fuzzy matrix A, denoted by adj A, is defined as follows: b0 = IAjil where tAj~I is the determinant of the (n - 1) x (n - 1) fuzzy matrix formed by deleting row j and column i from A and B = adj A.
Remark 1. We note that IAjil can be obtained from IAI by replacing aji by 1 and all other row-]" factors ajk, k ~ i, by 0. We also can rewrite the element b~j of adj A as
o- ~Snjn~ t~n)
where nj = {1, 2 . . . . .
n}',{j} and Sn,n, is the set of all permutations of set nj over the set ni.
300
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
Proposition 4.2. For n × n f u z z y matrices A and B we have the following: (1) A ~ B implies adj A ~
Z ]--I at~r(t) and ~ ~s.j~i t ~ I
dij =
Z ~I b,,(,). ~ ~s~j.i ten,
It is clear that cq ~
Z
I~ a,¢(,)
and
cij =
Z
l~
a,¢(,)
which is the element bji. Hence (adj A)' = adj A' Proposition 4.3. Let A be an n x n f u z z y matrix. Then (1) A adj A >/IAI I,, (2) (adj A ) A >i IAI In. Proof. (1) Let C -- A adj A. The i-th row of A is (ai,, ai2 . . . . .
ai,).
By definition of adj A, the j-th column of adj A is given by (IAjll, I m j 2 [ , . . . , IAjnl).' So that cq = ~ aik IAjkl >>-0 k-I
and hence Cii ~---~
aik IAikl
k=l
which is equal to IAI. Thus C = A adj A/> IAI/,. (2) The proof is similar to (1). Proposition 4.4. I f a f u z z y matrix A has a zero row then (adj A ) A = 0 (the zero matrix). Proof. Let G = (adj A ) A . That is, gij = Z Imkil akj. k
If the i-th row of A is zero, then Ak~ contains a zero row where k ~ i and so IAk~l= 0 (by Proposition 3.3) for every k # i and if k = i, then aq = 0 for every j and hence Z IAki[ akj = O. k
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
301
Thus (adj A )A = O. T h e o r e m 4.5. For a fuzzy matrix A we have IAI = ladj ml.
Proof. Since
IAl,I adj A =
IA~,I "'" IA.11] • "" IA..~I ~
IA'l~l IZe, I • -"
IA~I_I
we have ladj AI = ~] IAI,.~I IA2~(2)1 • • • IA,~,)I ¢r E S n
=
~]
n IA,~(,>I
= ~]
~r E S n i = 1
= ~] [ (
OE~nn
[)(
I-I ato(,))(
~'~
O-~n
o" ~ S n
/
(I) t ~ n ]
I ] a,o(,) O" E Snin~(i } l E n t
~
)]
,~ a,o(,))'"(o~s~
OESn2~(2)
-
= ~so[(~.,a'°,(t))(,~2at°~(")"'(~.a'°.(t') (for some O~ E S.,.~,,). 02 ~ S.~..(2, . . . . .
' l-I a,o(,,)] n ~n
tegln
] O. ~ S.°.~(.,))
= ~ [(azo,(e)a30,(3)'"a.Ol(.))(alo~(,)a302(3)"" a . O z ( n ) ) ' " (a,oo(,)azoo(z)'"a.,-,o,,~.-,))] cr~Sn
= ~. [(a10~(l)alo~(l>"
" aloo~l))(a20,(z>a20~z>'"azo.~z))(a~o,~3)a30~o)a~o.(3>'"a30.,~3>)""
~rESn
X
(a,o,(ma,o~(m...a,o. ,(m)]
= ~ [(alo~,o)a2oiM)'"a~oi.(,))] for some f~ E {1, 2 . . . . . n}\{h}, h = 1, 2 . . . . . n. But since
ahoj~(~) ~ ah~.(f~), we have a~oj~(h)= ah~.(~. Therefore ladj AI = ~
a,~(,)a2~(2)" • • an~(.,)
cT ~ S n
which is the expansion of )AI. This completes the proof.
Proposition 4.6. Let A be an n x n constant fuzzy matrix. Then we have: (1) (adj A)' is constant, (2) C = A(adj A) is constant and C~j = Iml which is the least element in A. Proof. Let B = adj A. Then
big = Z
H a..(,)
o" c S n l n i t e n /
and
bik Z
H a..u).
o-Egnkni t ~ n k
We notice that bij = bi~ since the numbers o-(t) of columns cannot be changed in the two expansions of bij and b~k. So that (adj A)' is constant.
M.Z. Ragab, E.G. E m a m / The determinant and adjoint o f a square f u z z y matrix
302
(2) Since A is constant we can see that Aj~ cij = ~ aik IAikl-- 2 k=l
=
Aik and so IAj~I = IAikl for every i, j e {1, 2 . . . . .
n}. Thus
aik IAi~l = IAI.
k=l
NOW,
IAI = Z o-~S~
al~r(1)a2¢(2)
o
o
i
ariz.(n)
=
alo-(n)
=
a2o-(2)
• " "
amr(n)
for any t r e S, (since A is constant). Taking tr the identity p e r m u t a t i o n we get IAI = a11a22 • • • a , , which is the least element in A.
Proposition 4.7 [5]. Let A be an n x n reflexive f u z z y matrix. Then adj A = A c where A c is idempotent and c <~n - 1.
Proposition 4.8 [5]. I f A <-B, then A C <~BC. Proposition 4.9. Let A be an n x n reflexive f u z z y matrix. Then we have the following: (1) adj A 2 = (adj A) 2 = adj A, (2) if A is idempotent, then adj A = A, (3) adj A is reflexive, (4) adj(adj A) = adj A, (5) adj A ~>A, (6) A(adj A) = (adj A ) A = adj A. Proof. (1) Since A is reflexive, we get A 2 is also reflexive and adj A 2 = ( A 2 ) c = ( A C ) 2 = (adj A ) 2. But since A C is idempotent, we have (adj A) z = adj A. (2) We have by Proposition 4.7 that adj A = A C (c ~< n - 1). But we have also that A is idempotent. So A ~ = A. Thus adj A = A. (3) Let B = adj A. T h a t is, ~ atcr(t ).
b, = ~,
o" ESni t e n t
Taking the identity p e r m u t a t i o n tr(t) = t we get bii ~
alia22
" " "
a(i-1)(i-1)a(i+ l)(i+l) • " • ann =
1,
bii = 1 and adj A is thus reflexive. (4) Since A is reflexive, we get adj A is i d e m p o t e n t by the above Proposition and reflexive by (3). So that by (2) adj(adj A) = adj A. (5) Let B = adj A. T h a t is i.e.,
bij =
~
1-[ a,~(,).
¢2rESnjni t e n I
Taking the p e r m u t a t i o n tr(h)= h, tr(i)= j, h ~ i, i.e., the p e r m u t a t i o n 1
2
3
...
i
...
j-1
j+l
...
n~
1
2
3
--.
j
...
j-1
j+l
...
n
)
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
303
then a i la22a33
" " •
aija(j_ ~)(j_ 1)au+ ~)(~+~) • • •
ann
is a term of b~j. So that b~j >1a l l a 2 2 a 3 3
• • •
a~ja(j ~)(j_~)atj+~)(j+~) • • • a n n
=
a~j.
Therefore B = adj A/> A. (6) Let C = A(adj A) and D = (adj A ) A . Then cij = ~ a~k IAj~]/> a i i Imjil =
[Ajit
:
bij
k-I
and dij = ~ IAkil akj >~ IAjil ajj = k=l
IAjsl =
bo.
Thus we have A(adj A)/> adj A and (adj A ) A >! adj A. But by (1) and (5) and Proposition 4.8 we see that adj A = (adj A)(adj A) ~>A adj A. So that A(adj A) = adj A. Also adj A = (adj A)(adj A) ~ (adj A ) A so that (adj A ) A = adj A. Thus we get A(adj A) = (adj A ) A = adj A.
Example 2. Let
A =
[
]
1.0 0.3
0.5 1.0
0.8 0.4 .
0.7
0.6
1.0
Then All=
1.0 0.6
0.4 1.0 =1"0'
z,2= ~:~ lo =o4
0.3 A13 = 0.7
1.0 0.6 = 0.7,
A21 =
°0:~ ~ 08=06, 1.0
1.00.81.0 =1'0' A22= 0.7
1.0 A23 = 0.7
0.5 0.6 = 0.6,
0.8 0.4 =0"8'
1 . 0 0 . 8 =0.4, A32= 0.3 0.4
A33 =
0.5 A31= 1.0 Then
0.4
lol3O o~_1o 1.0
.11 A2 L,o 06 08] adjA=/A,2 A321= A22
LAI3
A23
0.41.00.4, 0.7 0.6 1.0
A33_]
A(adjA)=
I'.° 04°8][10: [,0060o41 : o4
(adj .4).4 =
[lOO6o8] 11o 1.oo4 o81[o4
0.3 1.0 0.70.61.0 0.4 0.7
1.0 0.6
0.4 1.0 0.70.61.0
0.4 1.0
=
0.3 0.7
1.0 0.6
0.7
0.6 1.0 0.6
0.8] 0.4, 1.oA
0.7
0.6 1.0 0.6
0.8 ] 0.4 . 1.0
=
1.0
It is clear that this example satisfies the statements of the above Proposition.
304
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
Definition 4.10. An n × n fuzzy matrix A is called circular if and only if (A2) ' < A, or more explicitly, ajkaki <~aij for every k = 1, 2 , . . . , n. Theorem 4.11. For an n × n fuzzy matrix A we have the following: (1) if A is symmetric, then adj A is symmetric, (2) if A is transitive, then adj A is transitive, (3) if A is circular, then adj A is circular. Proof. (1) Let B = adj A. Then bij =
~ I-[ atcr(t)= o-ESnjnit~nj
~
I--I a,r(t)t= bit
o" ESnin] t ~ n i
(since A is symmetric). (2) Let D = Aij. We can determine the elements of D in terms of the elements of A as follow: I ahk = ~a(n+l)k
d~k
if h < i, k < j, ifh>~i,k
I
if h < i, k ~>j,
ka(~+l)(k+l)
if h/> i, k ~>j,
where A~j denotes the ( n - 1 ) × ( n - 1) fuzzy matrix obtained from A by deleting the i-th row and column j. Now we show that AstAtu <<-As,, for every t E {1, 2 . . . . . n}. Let R =As,, C = A , , , , F =As,, and W = AstAt,,. Note that A is transitive. Then n
1
w 0 = ~ rikCkj k=l
n 1
=S, aikakj <<-ai~ = fi~
if i < s, k < t, j < u,
k=l n
1
= ~ aikak(y+l)<~ai(i+o=fij i f i < s , k < t , j>~u, k-1 n-1 = ~ ai(k+l)a(k+l)j<~aij=fij i f i < s , k>-t, j < u , k-1 n 1 = ~ ai(k+lla(k+l)(j+l)<~ai(]+l)=fiij i f i < s , k>~t, j>~u, k-1 n-1 = ~ a(i+l)kakj~a(i+l)i=fiy ifi>~s, k < t , j < u , k=l n-1
= ~ a(i+l)(k+l)a(k+l)y~a(i+l)]=fij
ifi>~s, k>~t,
j
k=l n-I
= E a(i+l)(k+l)a(k+l)(j+l)<~a(i+l)(j+l)=fi] ifi>~s, k>~t, j < u , k=l n-1 = E a(i+l)kak(j+D<~a(i+D(/+D=fiij i f i > - s , k < t , j>~u. k-I
Thus wij <~fq in every case and therefore As,Atu <<-As, for every t ~ {1, 2 . . . . . n}. By Theorem 3.4 we get
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
This means bt~b,, <- bus, i.e., b,,b,s <~b,~ for every t ~ {1, 2 . . . . . n}. Hence B = adj A is transitive. (3) Similarly, as in (2) we can show that A,,A,, <<-A,',s for every t E {1, 2 . . . . . n} so that LA,,I LA,,I ~ IA,,~l = IA,,I
.
Thus b,.,b,, <~b,,,. and B = adj A is circular. This completes the proof of the theorem.
Corollary 4.12. If a fuzzy matrix A is similarity then adj A is also similarity. Example 3. Let A be the fuzzy matrix 0.6 0.5 0.6 0.6 Then
0.7 0.6 0.7 0.6
0.6 0.5 0.6 0.6
0.6 1 0.5 0.6" 0.6
E0607 06 061r0607 06 06] [06 A2=
05
06
05
o5//o5
0.6 0.7 0.6 0 . 6 / / 0 . 6 0.6 0.6 0.6 0 . 6 / L 0 . 6
05
05
0.7 0.6 0.6 0.6 0.6 0.6
=
0.6 0.6 0.6 0.6 0.6 0.6 05
0.6 0.5 0.6 0.6
0.6] 0.5 ~
i.e., A is transitive. Now we calculate the adjoint of A as follows: 0.6 0.5 0.5 I IA.I = 0.7 0.6 0.6 =0.6, 0.6 0.6 0.6
0.5 0.5 0.5 IA121= 0.6 0.6 0.6 ==0.5, 0.6 0.6 0.6
0.5 0.6 0.5 IA~31= 0.6 0.7 0.6 =0.6, 0.6 0.6 0.6
0.5 0.6 0.5 IA14I = 0.6 0.7 0.6 =0.6, 0.6 0.6 0.6
0.7 0.6 0.6 IA21I = 0.7 0.6 0.6 =0.6, 0.6 0.6 0.6
0.6 0.6 0.6 IA221= 0.6 0.6 0.6 =0.6, 0.6 0.6 0.6
0.6 0.7 0.6 I 0.6 0.7 0.6 =0.6, 1A231= 0.6 0.6 0.6
0.6 0.7 0.6 IAz41= 0.6 0.7 0.6 =0.6, 0.6 0.6 0.6
0.7 0.6 0.6 IA311= 0.6 0.5 0.5 =0.6, 0.6 0.6 0.6
0.6 0.6 0.6 I IA321= 0.5 0.5 0.5 =0.5, 0.6 0.6 0.6
0.6 0.7 0.6 IA331= 0.5 0.6 0.5 =0.6, 0.6 0.6 0.6
1A341 =
0.6 0.7 0.6 0.5 0.6 0.5 =0.6, 0.6 0.6 0.6
0.7 0.6 0.6 IA41l = 0.6 0.5 0.5 =0.6, 0.7 0.6 0.6
IA421 =
Im431 =
0.6 0.7 0.5 0.6 06 0.5 I =0.6, 0.6 0.7 0.6
IA441 =
0.6 0.7 0.6 I 0.5 0.6 0.5 =0.6. 0.6 0.7 0.6
0.6 0.6 0.6 0.5 0.5 0.5 =0.5, 0.7 0.6 0.6
305
306
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
Then
a~A=
0.6
0.6
0.6
0.6]
0.5 0.6 0.6
0.6 0.6 0.6
0.5 0.6 0.6
0.5 0.6 0.6
[0.6 0.6 0.6 0.61r0.60.. .6 0.6 0.61:[0.60.6 0.6 0.6] 0.5 0.6 0.6 0.6 0.6 0.6
(adj A) 2=
0.5 0 . 5 1 1 0 . 5 0.6 0 . 6 / 1 0 . 6 0.6 0 . 6 J L 0 . 6
0.6 0.6 0.6
0.5 0.6 0.6
0.5 0.6 0.6
0.5 0.6 0.6
0.6 0.5 0.5 0.6 0.6 0.6 0.6 0.6 0.6
~< adj A,
i.e., adjA is transitive. It is easy to see that AikAkj<~Aq for every i , j , k ~{1,2,3,4}. Note that IAI -- ladj AI -- 0.6. Although the following theorem has an obvious proof, it is important. It enables us to construct a transitive fuzzy matrix from a given one. It is useful for studying traansitive fuzzy matrices or transitive fuzzy relations.
Theorem 4.13. For any n x n fuzzy matrix A, the fuzzy matrix A adj A is transitive.
Proof. Let C - - A adj A, i.e., aik IAjk] = aiflAjf]
Cij = ~ k=l
for some f E {1, 2 , . . . , n}, and C}2) =
CisCsj= s=l
ail s=l
as, IAj,[
lAst]
=
t=l
= 2 aih IAshl as, IAi, I ~
Example 4. Let
A =
[0.50.7]0.8 0.3 0.9
0.6 0.2
and ( A a d j A ) 2=
0.4 . 1.0
Then
adjA=
[0.60.70.6] 0.4 0.6
0.8 0.7
0.4 0.5
i06 07 0.5][0.607 051 i06 06 051 0.4 0.6
0.6 0.7
0.4 0.6
0.4 0.6
0.6 0.7
0.4 0.6
=
0.4 0.6
0.6 0.6
0.4 0.6
~
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
307
Acknowledgement The authors are indebted to the referees for their helpful comments and suggestions.
References [1] Jin Bai Kim, Inverses of Boolean matrices, Bull. Inst. Math. Acad. Sinica 12 (1984) 125-128. [2] Jin Bai Kim, Determinant theory for fuzzy and Boolean matrices, Congressus Numerantium (1988) 273--276, Utilitas Mathematica pub. [3] Jin Bai Kim, Alphonse Baartmans and Nor Shakila Sahadin, Determinant theory for fuzzy matrices, Fuzzy Sets and Systems 29 (1989) 349-356. [4] Li Jian-Xin Controllable fuzzy matrices, Fuzzy Sets and Systems 45 (1992) 313-319. [5] M.G. Thomason, Convergence of powers of fuzzy matrix, J. of Math. Anal, Appl. 57 (1977) 476-480.
297
The determinant and adjoint of a square fuzzy matrix M.Z. Ragab and E.G. Emam Department of Mathematics, Zagazig University, Zagazig, Egypt Received March 1993 Revised July 1993
Abstract: The theory of the determinant and adjoint of a square fuzzy matrix will be studied. Also we define the circular fuzzy matrices and show that some properties of a square fuzzy matrix (such as reflexivity, transitivity, circularity) are carried over to the adjoint of the matrix. Finally we show how to construct a transitive fuzzy matrix from a given one through the adjoint matrix.
Keywords: Determinant of a square fuzzy matrix; adjoint of a square fuzzy matrix.
1. Introduction
There are some papers on the determinant theory of square fuzzy matrices (see Kim [2, 3]). In the present work we study and investigate some properties of the determinant of a square fuzzy matrix. However, the results are similar to the crisp case. The adjoint of a square fuzzy matrix is defined by Thomason [5] and Kim [3]. We state a formula for the adjoint matrix (see Remark 1) and use this formula anywhere in the paper. In this paper we define the circular fuzzy matrices and show that some properties of a square fuzzy matrix such as reflexivity, transitivity and circularity are carried over to the adjoint matrix and we establish some results including that A adj A/> JA] I, (adj A ) A >i IAI I and IAI = ladj AI where IAI denotes the determinant of A and adj A denotes the adjoint matrix of A. At the end of this paper we prove that A adj A is transitive for any square fuzzy matrix A. However this result enables us to construct a transitive fuzzy matrix from a given one through the adjoint matrix. 2. Preliminaries
We give here some definitions which are applied in the paper. Note that the elements of a fuzzy matrix takes its values from the unit interval [0, 1]. Definition 2.1 [4,5]. For fuzzy matrices A = [aij] (n x n ) , B = [bij] (n × p ) and C = [qj] (n × p ) , the following operations are defined
B + C = [bij + cij] where bit + cij = max(b/h qj), A B = I ~ jl aikbkJ =
where aikbkj = min(a~, bkJ) ,
A ' - - [aj~] (the transpose of A),
A ~ = la~v~'], A k-~' = A k A (k = O, 1, 2 . . . . ), A ° = I,, where In is the usual identity matrix, B<~C i f f b ~ j ~ < q j f o r a l l i ~ { 1 , 2 . . . . . n } a n d j ~ { 1 , 2 , . . . , p } . Correspondence to: Dr. M.Z. Ragab, Department of Mathematics, Zagazig University, Zagazig, Egypt. 0165-0114/94/$07.00 © 1994~Elsevier Science B.V. All rights reserved SSDI: 0165-0114(93)E0188-X
M.Z. Ragab, E.G. Emam
298
/ The determinant and adjoint of a square fuzzy
matrix
Definition 2.2 [4]. L e t A be an n / n fuzzy matrix. T h e n A is said to be s y m m e t r i c iff A = A ' , reflexive iff A / > In, transitive iff A 2 ~< A, similarity iff it is reflexive, s y m m e t r i c and transitive, and i d e m p o t e n t iff AZ=A.
Definition 2.3. An m × n fuzzy matrix A is said to be constant if aik equal to each other.
=
ajk for all
i, j, k,, i.e., its rows are
3. The determinant of a square fuzzy matrix
Definition 3.1 [3]. T h e d e t e r m i n a n t IAI of an n z n fuzzy matrix A is defined as follows: IAI = ~
al~(1)az,~(2)"'an,~(n)
o'ESn
w h e r e Sn d e n o t e s the s y m m e t r i c g r o u p of all p e r m u t a t i o n s of the indices (1, 2 , . . . , n).
Proposition 3.2. I f a f u z z y matrix B is obtained f r o m an n × n f u z z y matrix A by multiplying the i-th row o f A (i-th c o l u m n ) by k ~ (0, 1], then k IAI = Inl.
Proof. By definition IBI-- ~
blo-(1)b2o-(2)"
""
bn,,(,,)
o-~Sn
=
~
alo-(1)a2tr(2)
• • •
kaio-(i) • • • am~(n )
~,ES n
=
k
Z
al~(1)a2~(2)
" " "
a i d ( i ) " " " anon(n)
o- ~ S n
= k IAI
Proposition 3.3. Let A be an n x n f u z z y matrix. Then we have: (1) if A contains a zero row (column), then IAI = 0, n a (2) if A is triangular, then [AI = I-I~'=l aii where ]-Ik=l k = ala2"
• an.
Proof. (1) E a c h t e r m in IAI contains a factor of each row ( c o l u m n ) and hence contains a factor of zero row (column), so that each t e r m of IAI is equal to zero and c o n s e q u e n t l y IAI = 0. (2) S u p p o s e A = [au] is triangular f r o m below, i.e., a u = 0 for i < j . T a k e a t e r m t of IAI t =
alo-(1)a2o-(2
) • • •
ang(n).
Let o.(1) ~ 1, so that 1 < o.(1) and so a1,,(1) = 0 and t = 0. This m e a n s that each t e r m is zero if o.(1) ~ 1. Let n o w o-(1) = 1 but o'(2) ~ 2. T h e n 2 < o'(2) and a2~(2) = 0 and t = 0. This m e a n s that each t e r m is zero
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
299
if 0-(1) ¢ 1 or 0-(2) ~ 2. However, in a similar manner we can see that each term for which 0-(1) ¢ 1 or 0-(2) ~ 2 . . . or 0-(n)¢ n must be zero. Consequently
qAI
=
a l i a 2 2
"
"
"
ann = ~ I aii" i=l
From the above proposition it is clear that III= 1.
Theorem 3.4 [2]. L e t A and B be two f u z z y matrices. Then IABI >1 IZl IBI.
Example 1. For a fuzzy matrix
A=
0.5 0.6
0.3 0.2
0.8] 0.9
0.0
0.7
0.4
we calculate the determinant IAI as follows: 001~ JA[ =0.5
0.9 + 0.3 0.6 0.4 0.0
0.9 0.6 0.4 +0"8 0.0
0.2 0.7
= 0.5(0.2 + 0.7) + 0.3(0.4 + 0.0) + 0.8(0.6 + 0.0) = 0.5(0.7) + 0.3(0.4) + 0.8(0.6) = 0.5 + 0.3 + 0.6 = 0.6.
4. The adjoint of a square fuzzy matrix This is the principal section in this paper. We begin with the following.
Definition 4.1 [3]. The adjoint matrix of an n x n fuzzy matrix A, denoted by adj A, is defined as follows: b0 = IAjil where tAj~I is the determinant of the (n - 1) x (n - 1) fuzzy matrix formed by deleting row j and column i from A and B = adj A.
Remark 1. We note that IAjil can be obtained from IAI by replacing aji by 1 and all other row-]" factors ajk, k ~ i, by 0. We also can rewrite the element b~j of adj A as
o- ~Snjn~ t~n)
where nj = {1, 2 . . . . .
n}',{j} and Sn,n, is the set of all permutations of set nj over the set ni.
300
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
Proposition 4.2. For n × n f u z z y matrices A and B we have the following: (1) A ~ B implies adj A ~
Z ]--I at~r(t) and ~ ~s.j~i t ~ I
dij =
Z ~I b,,(,). ~ ~s~j.i ten,
It is clear that cq ~
Z
I~ a,¢(,)
and
cij =
Z
l~
a,¢(,)
which is the element bji. Hence (adj A)' = adj A' Proposition 4.3. Let A be an n x n f u z z y matrix. Then (1) A adj A >/IAI I,, (2) (adj A ) A >i IAI In. Proof. (1) Let C -- A adj A. The i-th row of A is (ai,, ai2 . . . . .
ai,).
By definition of adj A, the j-th column of adj A is given by (IAjll, I m j 2 [ , . . . , IAjnl).' So that cq = ~ aik IAjkl >>-0 k-I
and hence Cii ~---~
aik IAikl
k=l
which is equal to IAI. Thus C = A adj A/> IAI/,. (2) The proof is similar to (1). Proposition 4.4. I f a f u z z y matrix A has a zero row then (adj A ) A = 0 (the zero matrix). Proof. Let G = (adj A ) A . That is, gij = Z Imkil akj. k
If the i-th row of A is zero, then Ak~ contains a zero row where k ~ i and so IAk~l= 0 (by Proposition 3.3) for every k # i and if k = i, then aq = 0 for every j and hence Z IAki[ akj = O. k
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
301
Thus (adj A )A = O. T h e o r e m 4.5. For a fuzzy matrix A we have IAI = ladj ml.
Proof. Since
IAl,I adj A =
IA~,I "'" IA.11] • "" IA..~I ~
IA'l~l IZe, I • -"
IA~I_I
we have ladj AI = ~] IAI,.~I IA2~(2)1 • • • IA,~,)I ¢r E S n
=
~]
n IA,~(,>I
= ~]
~r E S n i = 1
= ~] [ (
OE~nn
[)(
I-I ato(,))(
~'~
O-~n
o" ~ S n
/
(I) t ~ n ]
I ] a,o(,) O" E Snin~(i } l E n t
~
)]
,~ a,o(,))'"(o~s~
OESn2~(2)
-
= ~so[(~.,a'°,(t))(,~2at°~(")"'(~.a'°.(t') (for some O~ E S.,.~,,). 02 ~ S.~..(2, . . . . .
' l-I a,o(,,)] n ~n
tegln
] O. ~ S.°.~(.,))
= ~ [(azo,(e)a30,(3)'"a.Ol(.))(alo~(,)a302(3)"" a . O z ( n ) ) ' " (a,oo(,)azoo(z)'"a.,-,o,,~.-,))] cr~Sn
= ~. [(a10~(l)alo~(l>"
" aloo~l))(a20,(z>a20~z>'"azo.~z))(a~o,~3)a30~o)a~o.(3>'"a30.,~3>)""
~rESn
X
(a,o,(ma,o~(m...a,o. ,(m)]
= ~ [(alo~,o)a2oiM)'"a~oi.(,))] for some f~ E {1, 2 . . . . . n}\{h}, h = 1, 2 . . . . . n. But since
ahoj~(~) ~ ah~.(f~), we have a~oj~(h)= ah~.(~. Therefore ladj AI = ~
a,~(,)a2~(2)" • • an~(.,)
cT ~ S n
which is the expansion of )AI. This completes the proof.
Proposition 4.6. Let A be an n x n constant fuzzy matrix. Then we have: (1) (adj A)' is constant, (2) C = A(adj A) is constant and C~j = Iml which is the least element in A. Proof. Let B = adj A. Then
big = Z
H a..(,)
o" c S n l n i t e n /
and
bik Z
H a..u).
o-Egnkni t ~ n k
We notice that bij = bi~ since the numbers o-(t) of columns cannot be changed in the two expansions of bij and b~k. So that (adj A)' is constant.
M.Z. Ragab, E.G. E m a m / The determinant and adjoint o f a square f u z z y matrix
302
(2) Since A is constant we can see that Aj~ cij = ~ aik IAikl-- 2 k=l
=
Aik and so IAj~I = IAikl for every i, j e {1, 2 . . . . .
n}. Thus
aik IAi~l = IAI.
k=l
NOW,
IAI = Z o-~S~
al~r(1)a2¢(2)
o
o
i
ariz.(n)
=
alo-(n)
=
a2o-(2)
• " "
amr(n)
for any t r e S, (since A is constant). Taking tr the identity p e r m u t a t i o n we get IAI = a11a22 • • • a , , which is the least element in A.
Proposition 4.7 [5]. Let A be an n x n reflexive f u z z y matrix. Then adj A = A c where A c is idempotent and c <~n - 1.
Proposition 4.8 [5]. I f A <-B, then A C <~BC. Proposition 4.9. Let A be an n x n reflexive f u z z y matrix. Then we have the following: (1) adj A 2 = (adj A) 2 = adj A, (2) if A is idempotent, then adj A = A, (3) adj A is reflexive, (4) adj(adj A) = adj A, (5) adj A ~>A, (6) A(adj A) = (adj A ) A = adj A. Proof. (1) Since A is reflexive, we get A 2 is also reflexive and adj A 2 = ( A 2 ) c = ( A C ) 2 = (adj A ) 2. But since A C is idempotent, we have (adj A) z = adj A. (2) We have by Proposition 4.7 that adj A = A C (c ~< n - 1). But we have also that A is idempotent. So A ~ = A. Thus adj A = A. (3) Let B = adj A. T h a t is, ~ atcr(t ).
b, = ~,
o" ESni t e n t
Taking the identity p e r m u t a t i o n tr(t) = t we get bii ~
alia22
" " "
a(i-1)(i-1)a(i+ l)(i+l) • " • ann =
1,
bii = 1 and adj A is thus reflexive. (4) Since A is reflexive, we get adj A is i d e m p o t e n t by the above Proposition and reflexive by (3). So that by (2) adj(adj A) = adj A. (5) Let B = adj A. T h a t is i.e.,
bij =
~
1-[ a,~(,).
¢2rESnjni t e n I
Taking the p e r m u t a t i o n tr(h)= h, tr(i)= j, h ~ i, i.e., the p e r m u t a t i o n 1
2
3
...
i
...
j-1
j+l
...
n~
1
2
3
--.
j
...
j-1
j+l
...
n
)
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
303
then a i la22a33
" " •
aija(j_ ~)(j_ 1)au+ ~)(~+~) • • •
ann
is a term of b~j. So that b~j >1a l l a 2 2 a 3 3
• • •
a~ja(j ~)(j_~)atj+~)(j+~) • • • a n n
=
a~j.
Therefore B = adj A/> A. (6) Let C = A(adj A) and D = (adj A ) A . Then cij = ~ a~k IAj~]/> a i i Imjil =
[Ajit
:
bij
k-I
and dij = ~ IAkil akj >~ IAjil ajj = k=l
IAjsl =
bo.
Thus we have A(adj A)/> adj A and (adj A ) A >! adj A. But by (1) and (5) and Proposition 4.8 we see that adj A = (adj A)(adj A) ~>A adj A. So that A(adj A) = adj A. Also adj A = (adj A)(adj A) ~ (adj A ) A so that (adj A ) A = adj A. Thus we get A(adj A) = (adj A ) A = adj A.
Example 2. Let
A =
[
]
1.0 0.3
0.5 1.0
0.8 0.4 .
0.7
0.6
1.0
Then All=
1.0 0.6
0.4 1.0 =1"0'
z,2= ~:~ lo =o4
0.3 A13 = 0.7
1.0 0.6 = 0.7,
A21 =
°0:~ ~ 08=06, 1.0
1.00.81.0 =1'0' A22= 0.7
1.0 A23 = 0.7
0.5 0.6 = 0.6,
0.8 0.4 =0"8'
1 . 0 0 . 8 =0.4, A32= 0.3 0.4
A33 =
0.5 A31= 1.0 Then
0.4
lol3O o~_1o 1.0
.11 A2 L,o 06 08] adjA=/A,2 A321= A22
LAI3
A23
0.41.00.4, 0.7 0.6 1.0
A33_]
A(adjA)=
I'.° 04°8][10: [,0060o41 : o4
(adj .4).4 =
[lOO6o8] 11o 1.oo4 o81[o4
0.3 1.0 0.70.61.0 0.4 0.7
1.0 0.6
0.4 1.0 0.70.61.0
0.4 1.0
=
0.3 0.7
1.0 0.6
0.7
0.6 1.0 0.6
0.8] 0.4, 1.oA
0.7
0.6 1.0 0.6
0.8 ] 0.4 . 1.0
=
1.0
It is clear that this example satisfies the statements of the above Proposition.
304
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
Definition 4.10. An n × n fuzzy matrix A is called circular if and only if (A2) ' < A, or more explicitly, ajkaki <~aij for every k = 1, 2 , . . . , n. Theorem 4.11. For an n × n fuzzy matrix A we have the following: (1) if A is symmetric, then adj A is symmetric, (2) if A is transitive, then adj A is transitive, (3) if A is circular, then adj A is circular. Proof. (1) Let B = adj A. Then bij =
~ I-[ atcr(t)= o-ESnjnit~nj
~
I--I a,r(t)t= bit
o" ESnin] t ~ n i
(since A is symmetric). (2) Let D = Aij. We can determine the elements of D in terms of the elements of A as follow: I ahk = ~a(n+l)k
d~k
if h < i, k < j, ifh>~i,k
I
if h < i, k ~>j,
ka(~+l)(k+l)
if h/> i, k ~>j,
where A~j denotes the ( n - 1 ) × ( n - 1) fuzzy matrix obtained from A by deleting the i-th row and column j. Now we show that AstAtu <<-As,, for every t E {1, 2 . . . . . n}. Let R =As,, C = A , , , , F =As,, and W = AstAt,,. Note that A is transitive. Then n
1
w 0 = ~ rikCkj k=l
n 1
=S, aikakj <<-ai~ = fi~
if i < s, k < t, j < u,
k=l n
1
= ~ aikak(y+l)<~ai(i+o=fij i f i < s , k < t , j>~u, k-1 n-1 = ~ ai(k+l)a(k+l)j<~aij=fij i f i < s , k>-t, j < u , k-1 n 1 = ~ ai(k+lla(k+l)(j+l)<~ai(]+l)=fiij i f i < s , k>~t, j>~u, k-1 n-1 = ~ a(i+l)kakj~a(i+l)i=fiy ifi>~s, k < t , j < u , k=l n-1
= ~ a(i+l)(k+l)a(k+l)y~a(i+l)]=fij
ifi>~s, k>~t,
j
k=l n-I
= E a(i+l)(k+l)a(k+l)(j+l)<~a(i+l)(j+l)=fi] ifi>~s, k>~t, j < u , k=l n-1 = E a(i+l)kak(j+D<~a(i+D(/+D=fiij i f i > - s , k < t , j>~u. k-I
Thus wij <~fq in every case and therefore As,Atu <<-As, for every t ~ {1, 2 . . . . . n}. By Theorem 3.4 we get
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
This means bt~b,, <- bus, i.e., b,,b,s <~b,~ for every t ~ {1, 2 . . . . . n}. Hence B = adj A is transitive. (3) Similarly, as in (2) we can show that A,,A,, <<-A,',s for every t E {1, 2 . . . . . n} so that LA,,I LA,,I ~ IA,,~l = IA,,I
.
Thus b,.,b,, <~b,,,. and B = adj A is circular. This completes the proof of the theorem.
Corollary 4.12. If a fuzzy matrix A is similarity then adj A is also similarity. Example 3. Let A be the fuzzy matrix 0.6 0.5 0.6 0.6 Then
0.7 0.6 0.7 0.6
0.6 0.5 0.6 0.6
0.6 1 0.5 0.6" 0.6
E0607 06 061r0607 06 06] [06 A2=
05
06
05
o5//o5
0.6 0.7 0.6 0 . 6 / / 0 . 6 0.6 0.6 0.6 0 . 6 / L 0 . 6
05
05
0.7 0.6 0.6 0.6 0.6 0.6
=
0.6 0.6 0.6 0.6 0.6 0.6 05
0.6 0.5 0.6 0.6
0.6] 0.5 ~
i.e., A is transitive. Now we calculate the adjoint of A as follows: 0.6 0.5 0.5 I IA.I = 0.7 0.6 0.6 =0.6, 0.6 0.6 0.6
0.5 0.5 0.5 IA121= 0.6 0.6 0.6 ==0.5, 0.6 0.6 0.6
0.5 0.6 0.5 IA~31= 0.6 0.7 0.6 =0.6, 0.6 0.6 0.6
0.5 0.6 0.5 IA14I = 0.6 0.7 0.6 =0.6, 0.6 0.6 0.6
0.7 0.6 0.6 IA21I = 0.7 0.6 0.6 =0.6, 0.6 0.6 0.6
0.6 0.6 0.6 IA221= 0.6 0.6 0.6 =0.6, 0.6 0.6 0.6
0.6 0.7 0.6 I 0.6 0.7 0.6 =0.6, 1A231= 0.6 0.6 0.6
0.6 0.7 0.6 IAz41= 0.6 0.7 0.6 =0.6, 0.6 0.6 0.6
0.7 0.6 0.6 IA311= 0.6 0.5 0.5 =0.6, 0.6 0.6 0.6
0.6 0.6 0.6 I IA321= 0.5 0.5 0.5 =0.5, 0.6 0.6 0.6
0.6 0.7 0.6 IA331= 0.5 0.6 0.5 =0.6, 0.6 0.6 0.6
1A341 =
0.6 0.7 0.6 0.5 0.6 0.5 =0.6, 0.6 0.6 0.6
0.7 0.6 0.6 IA41l = 0.6 0.5 0.5 =0.6, 0.7 0.6 0.6
IA421 =
Im431 =
0.6 0.7 0.5 0.6 06 0.5 I =0.6, 0.6 0.7 0.6
IA441 =
0.6 0.7 0.6 I 0.5 0.6 0.5 =0.6. 0.6 0.7 0.6
0.6 0.6 0.6 0.5 0.5 0.5 =0.5, 0.7 0.6 0.6
305
306
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
Then
a~A=
0.6
0.6
0.6
0.6]
0.5 0.6 0.6
0.6 0.6 0.6
0.5 0.6 0.6
0.5 0.6 0.6
[0.6 0.6 0.6 0.61r0.60.. .6 0.6 0.61:[0.60.6 0.6 0.6] 0.5 0.6 0.6 0.6 0.6 0.6
(adj A) 2=
0.5 0 . 5 1 1 0 . 5 0.6 0 . 6 / 1 0 . 6 0.6 0 . 6 J L 0 . 6
0.6 0.6 0.6
0.5 0.6 0.6
0.5 0.6 0.6
0.5 0.6 0.6
0.6 0.5 0.5 0.6 0.6 0.6 0.6 0.6 0.6
~< adj A,
i.e., adjA is transitive. It is easy to see that AikAkj<~Aq for every i , j , k ~{1,2,3,4}. Note that IAI -- ladj AI -- 0.6. Although the following theorem has an obvious proof, it is important. It enables us to construct a transitive fuzzy matrix from a given one. It is useful for studying traansitive fuzzy matrices or transitive fuzzy relations.
Theorem 4.13. For any n x n fuzzy matrix A, the fuzzy matrix A adj A is transitive.
Proof. Let C - - A adj A, i.e., aik IAjk] = aiflAjf]
Cij = ~ k=l
for some f E {1, 2 , . . . , n}, and C}2) =
CisCsj= s=l
ail s=l
as, IAj,[
lAst]
=
t=l
= 2 aih IAshl as, IAi, I ~
Example 4. Let
A =
[0.50.7]0.8 0.3 0.9
0.6 0.2
and ( A a d j A ) 2=
0.4 . 1.0
Then
adjA=
[0.60.70.6] 0.4 0.6
0.8 0.7
0.4 0.5
i06 07 0.5][0.607 051 i06 06 051 0.4 0.6
0.6 0.7
0.4 0.6
0.4 0.6
0.6 0.7
0.4 0.6
=
0.4 0.6
0.6 0.6
0.4 0.6
~
M.Z. Ragab, E.G. Emam / The determinant and adjoint of a square fuzzy matrix
307
Acknowledgement The authors are indebted to the referees for their helpful comments and suggestions.
References [1] Jin Bai Kim, Inverses of Boolean matrices, Bull. Inst. Math. Acad. Sinica 12 (1984) 125-128. [2] Jin Bai Kim, Determinant theory for fuzzy and Boolean matrices, Congressus Numerantium (1988) 273--276, Utilitas Mathematica pub. [3] Jin Bai Kim, Alphonse Baartmans and Nor Shakila Sahadin, Determinant theory for fuzzy matrices, Fuzzy Sets and Systems 29 (1989) 349-356. [4] Li Jian-Xin Controllable fuzzy matrices, Fuzzy Sets and Systems 45 (1992) 313-319. [5] M.G. Thomason, Convergence of powers of fuzzy matrix, J. of Math. Anal, Appl. 57 (1977) 476-480.