ON REGULARITY OF BLOCK TRIANGULAR FUZZY MATRICES 1

J. Appl. Math. & Computing Vol. 15(2004), No. 1 - 2, pp. 207 - 220

ON REGULARITY OF BLOCK TRIANGULAR FUZZY MATRICES AR. MEENAKSHI

Abstract. Necessary and sufficient conditions are given for the regularity of block triangular fuzzy matrices. This leads to characterization of idempotency of a class of triangular Toeplitz matrices. As an application, the existence of group inverse of a block triangular fuzzy matrix is discussed. Equivalent conditions for a regular block triangular fuzzy matrix to be expressed as a sum of regular block fuzzy matrices is derived. Further, fuzzy relational equations consistency is studied. AMS Mathematics Subject Classification : 15A26, 15A09 Key words and phrases : Fuzzy matrix, fuzzy relational equation, block triangular matrix, triangular Toeplitz matrix.

1. Introduction Let F[0, 1] be fuzzy algebra over the support [0, 1] with operations + and ·, defined as a + b = max{a, b} and a · b = min{a, b} for all a, b ∈ [0, 1] and the standard order ≥ . Let Fmn be the set of all m × n fuzzy matrices over F. A matrix A ∈ Fmn is said to be regular if there exists X ∈ Fnm such that AXA = A. In this case X is called a generalized inverse of A. Each element a ∈ F is regular, because axa = a holds under the fuzzy multiplication for all x ≥ a. Hence F is regular. It is well known that for arbitary ring R, R is regular if and only if Rmn , the set of all m × n matrices over R, is regular. However Received June 12, 2003. Revised December 10, 2003. c 2004 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.

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this fails for fuzzy matrices. This motivates the study on regularity of fuzzy matrices. In [3], an algorithm for a fuzzy matrix to be regular is given. Finite fuzzy relational equations can be expressed in the form of fuzzy matrix equation as x · A = b for some fuzzy coefficient matrix A ∈ Fmn and b ∈ F1n . In [1], the solution of a fuzzy matrix equation whose coefficient matrix is regular has been discussed. If A is regular with a generalized inverse X, then b · X is a solution of x · A = b. Further, every invertible matrix is regular. Regular fuzzy matrices play an important role in estimation and inverse problem in fuzzy relational equations [6, pp. 1-14] and in fuzzy optimization problems [5, pp. 533-539]. In Fuzzy Retrival system, the degree of relevance of the concept matrix depends on that of its transitive closure which is a regular matrix [2, pp. 691-714]. In this paper, we investigate the regularity of block triangular fuzzy matrices of the form     A B A 0 or , C D 0 D where the entries of each block are from the support [0, 1] of fuzzy algebra F. Since a matrix A is regular if and only if its transpose AT is regular and the transpose of a lower block triangular matrix is an upper block triangular matrix, throughout we shall only investigate the case of a lower block triangular matrix. In section 2, some basic definitions and results required are given. In section 3, equivalent conditions for regularity of block triangular fuzzy matrix are determined. The idempotency of block triangular fuzzy matrix is deduced as a corollary. As an application, the existence of group inverse of block triangular fuzzy matrix is discussed. In section 4, equivalent conditions for regular block triangular fuzzy matrix to be expressed as sum of regular block fuzzy matrices is derived. In section 5, consistency of fuzzy matrix equation x · M = b where M is a lower block triangular fuzzy matrix is studied. In section 6, characterization of idempotency of a class of triangular Toeplitz matrices is obtained.

2. Preliminaries For A ∈ Fmn , the row space R(A) is a subspace of F1n generated by the rows of A and the column space C(A) is defined in dual fashion. The row (column) rank of a fuzzy matrix A is the smallest possible size of a spanning set for R(A) (C(A) respectively). Let ρr (A) and ρc (A) denote the row rank and column rank of A, respectively. In general, familiar rank properties of field based matrices do not hold for fuzzy matrices. It has been proved in [3], that row and column rank of regular

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fuzzy matrix are equal. Hence, let us denote the rank of a regular matrix A as ρ(A). Definition 1. The matrix A ∈ Fnn is said to be invertible if there exists X ∈ Fnn such that A · X = X · A = In , where In stands for the identity matrix of order n. It is well known that (Corollary 1 of [7]) a fuzzy matrix is invertible if and only if it is a permutation matrix. Definition 2. The matrix A ∈ Fmn is called a regular matrix if there exists a matrix X ∈ Fnm , satisfying AXA = A. If AXA = A, X is called a generalized (g − ) inverse of A and is denoted as − A . If for the above X ∈ Fnm , the equality X A X = X also holds, then X is called a semi inverse of A and is denoted as A= . Definition 3. The Moore-Penrose inverse of the matrix A ∈ Fmn is a semi inverse X of A such that AX = (AX)T and XA = (XA)T . In the sequal, we make use the following Proposition 2.4 of [3]. Lemma 1. For A, B ∈ Fmn , the following statements hold: (i) R(B) ⊆ R(A) ⇔ there exists X ∈ Fmm such that B = XA. (ii) C(B) ⊆ C(A) ⇔ there exists Y ∈ Fnn such that B = AY.

3. Regular block triangular fuzzy matrices In this section, we derive equivalent conditions for regularity of block triangular fuzzy matrix of the form,   A 0 M= with R(C) ⊆ R(A) and C(C) ⊆ C(D). (1) C D First we prove certain lemmas that simplify the proofs of the main results. I denotes the identity matrix of appropriate size. Lemma 2. F or A, B ∈ Fmn ,if A is regular, then (i) R(B) ⊆ R(A) ⇔ B = BA− A for each A− of A, (ii) C(B) ⊆ C(A) ⇔ B = AA− B for each A− of A.

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Proof. According to Lemma 1, R(B) ⊆ R(A) ⇔ there exists X ∈ Fmm such that B = XA. By Definition 2, A = AA− A. Hence B = XA ⇒ B = X ·AA− A = BA− A. Conversely, suppose B = BA− A. Then we have B = XA, by taking X = BA− . Hence, R(B) ⊆ R(A) ⇔ B = BA− A. (ii) can be proved in the same mannar. 

Lemma 3. Let M =



A C

0 D



be a lower block triangular matrix. The follow-

ing statements hold: (i) If R(C) ⊆ R(A), then ρr (M ) = ρr (A) + ρr (D). (ii) If C(C) ⊆ C(D), then ρc (M ) = ρc (A) + ρc (D). Proof. From Lemma 1 it follows that if R(C) ⊆ R(A), then there exists X, such that C = XA. We express M in the form: 

where U =



I X

  0 M= I   0 A and L = I 0 I X

 A 0 = U L, 0 D  0 . ρr (M ) = ρr (U L) ≤ ρr (L) = D

ρr (A) + ρr (D).   A 0 Since M = , we get ρr (M ) ≥ ρr (A) + ρr (D). Hence ρr (M ) = C D ρr (A) + ρr (D). (ii) can be proved in the same manner.  Theorem 1. For any block triangular matrix M of the form (1), M is a regular matrix and M has a lower block triangular g-inverse if and only if the blocks A and D are regular matrices.   0 A− − Furthermore, ρ(M ) = ρ(A) + ρ(D) and M = . (2) D− CA− D− Proof. As in the proof of Lemma 3, M can be written as M = U L, where     I 0 A 0 U= and L = . X I 0 D Since A is regular, by Lemma 2, we have X = CA− . U is idempotent, that is, U 2 = U because X + X = X. Since U being idempotent, it is regular. U

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itself is choiceg-inverse for U . Since A and D are regular, L is regular.  one 0 A− − L = is a g-inverse of L. Thus both U and L are regular. 0 D− We claim that L− U is a g-inverse of M . Since R(C) ⊆ R(A) and C(C) ⊆ C(D), by Lemma 2, we get CA− A = C = DD− C. Now, using U 2 = U, we get M (L− U )M = U L(L− U )U L = M L− M. Now, −

M (L U )M =

 

A C

0 D



A− 0

0 D−

A = CA− A + DD− C   A 0 = C +C D   A 0 = = M. C D

0 D

 

A C

0 D



Hence M is regular. Thus, regularity of A and D implies that M is regular,   0 A− − − . with a lower block triangular g-inverse M = L U = D− CA− D− Conversely, suppose Mis regular and M has a lower block triangular g-inverse  X 0 and let M − = . Then M M − M = M yields AXA = A, DZD = D Y Z and CXA + CY A + DZC = C. From this it follows that A and D are regular. The rank equality ρ(M ) = ρ(A) + ρ(D) follows from Lemma 3.  Remark 1. Corollary 2 of [4] states that for a lower block triangular matrix   A 0 M = over a field with R(C) ⊆ R(A) (or C(C) ⊆ C(D)), M has a C D lower block triangular g-inverse. However for fuzzy matrices, both the conditions are essential. This is illustrated by the following example. Example 1. For the lower block triangular matrix   1 1 0 0  1 0 0 0   M =  0 0 1 1  1 1 1 0

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with A A A−



1 1 1 0 = D is regular,   0 1 = 1 α

=

D =



, C =



0 0 1 1



,

for α ∈ [0, 1] is a generalized inverse of A. C = CA− A but C 6= DD− C (D− = A− is a generalized inverse of D).   0 A− − is not Hence R(C) ⊆ R(A) holds and C(C) 6⊆ C(D). M = A− CA− D− a g-inverse of M. Remark 2. A lower block triangular matrix M with diagonal blocks that are regular may not be regular.   ·3 0 For instance, let M = , ·5 6= α(·3) for any α ∈ [0, 1], and ·5 6= (·2)β ·5 ·2 for any β ∈ [0, 1]. Hence both the contitions R(C) ⊆ R(A) and C(C) ⊆ C(D) fail. Both A and D are regular. Each row of M cannot be expressed as linear combination of the other row. Hence by Definition 2.5 of [3], the rows are linearly independent. By Definition 2.6 of [3], they form a standard basis for the row space of M.     1 0 0 1 For both permutation matrices P1 = and P2 = , 0 1 1 0     ·3 0 ·3 ·2 6= M and M P2 M = 6= M. M P1 M = ·3 ·2 ·5 ·2 Hence M is not regular by step 3 in Algorithm 1 of [3]. Namely, M is regular if and only if M P M = M for some permutation matrix P .  A 0 has a lower block triangular g-inverse M − of C D the form(2), then there exists a lower block triangular semi inverse for M of the   0 A= . form D= CA= D= Corollary 1. If M =



Proof. Since M − is a lower block triangular g-inverse of M, of the form (2) and M − M M − is a semi inverse of M, equating the corresponding blocks in M − M M − = M − we get A− AA− = A− and D− DD− = D− . Hence, A− and D− are semi

On regularity of block triangular fuzzy matrices

inverses of A and D respectively. Then  A= M= = = D CA=

0 D=

213



is a semi inverse of M .



It is clear that if M is regular, then it has a g-inverse and a semi inverse. For any fuzzy matrix, if the Moore-Penrose inverse exists, then by Theorm 3.16 of [3], it is unique and coincides with its transpose. Hence, a lower block triangular fuzzy matrix cannot have a lower block triangular Moore-Penrose inverse. Now, we define the group inverse of a fuzzy matrix and discuss the existence of the group inverse of a triangular block matrix in the following: Definition 4. A group inverse M # of a fuzzy matrix M is a semi inverse of M such that M M # = M # M. If M # exists, then it is unique. Theorem 2. Let M be of the form (1), A and D be square matrices. Then the group inverse M # exists if and only if the group inverses A# and D# exist and DC = CA. Proof. Since M has the form (1), it follows from Theorem 1 and Corollary 1 that     = 0 0 AA= A A = MM= = and M . M = CA= DD= D= C D= D According to the Definition 4 of group inverse if M # exists, then M M = = M = M. That is, AA= = A= A, DD= = D= D and CA= = D= C. It means that A# and D# exist and DC = D(CA= A) = D(CA= )A = D(D= CA) = (DD= C)A = CA. The converse follows by retracing the steps.



Corollary 2. Let M be of the form (1), A and D be square matrices. If M # exists, then ρ(M 2 ) = ρ(M ), ρ(A2 ) = ρ(A), ρ(D2 ) = ρ(D), ρ(M ) = ρ(A) + ρ(D).

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Proof. If M # exists, then M M = = M = M and M = M M =M = M 2 M =. Hence ρ(M ) = ρ(M 2 ). The rest of the proof follows from Theorem 2. Corollary 3. For the lower block triangular matrix M =



A C

 0 D



, the

following statements are equivalent: (i) M is idempotent. (ii) The blocks A and D are idempotent matrices and CA = DC = C. (iii) The blocks A and D are idempotent matrices with R(C) ⊆ R(A) and C(C) ⊆ C(D). Proof. (i)⇔(ii). M is idempotent ⇔ A and D are idempotent matrices and CA + DC = C. Suppose CA = DC = C. Then CA + DC = C + C = C. If CA + DC = C, then by fuzzy addition, C ≥ CA and C ≥ DC. we claim that CA = DC = C. For if C > CA, then CA > CA2 ⇒ CA > CA. If C > DC, then DC > D2 C ⇒ DC > DC. Both are not possible. (ii)⇔(iii). Since A and D are idempotent, we have A− = A and D− = D. Now, CA− A = CA2 = CA = C = DC = D2 C = DD− C. Thus by Lemma 2, CA = DC = C ⇔ R(C) ⊆ R(A) and C(C) ⊆ C(D).



4. Regular fuzzy matrix decomposition In this section, we prove that under certain conditions, a regular lower block triangular fuzzy matrix can be expressed as sum of two regular block fuzzy matrices. Lemma 4. For A, B ∈ Fmn ,  (i) [A B] is regular ⇔    A (ii) is regular ⇔ B

the following statements hold:    A B 0 0 is regular ⇔ is regular. 0 0 B A  A 0 is regular. B 0

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X Proof. Let [A B] be regular and be a g-inverse. By Definition 2, it follows Y   X that [A B] [A B] = [A B]. By equating the corresponding blocks on Y both sides, we get (AX + BY )A = A and (AX + BY )B = B.     X 0 A B On computation we see that is a g-inverse of . Thus Y 0 0 0   A B is regular. 0 0       X U A B A B Conversely, if is regular, let be a g-inverse of . Y V 0 0 0 0 Then again on computation we get (AX + BY )A = A and (AX + BY )B = B.   X This can be written as [A B] [A B] = [A B]. Thus [A B] is regular Y   A B ⇔ is regular. A fuzzy matrix M is regular ⇔ P M P T is regular for 0 0 a permutation matrix P, (By Definition 1, P is invertible). 

A 0

B 0





is regular ⇔ P  0 ⇔ B

A 0 0 A

B 0 



P T is regular

is regular.

(ii) can be proved in the same manner.



Lemma 5. For A, B, C ∈ Fmn , the following statements hold:   A (i) If R(C) ⊆ R(A), then A is regular ⇔ is regular. C (ii) If C(B) ⊆ C(A), then A is regular ⇔ [A B] is regular. Proof. (i) From Lemma 2, we get R(C) ⊆ R(A) and A is regular ⇒ C = CA− A.     A A − We see that [A 0] is a g-inverse of . Hence is regular. C C   A Conversely, if M = is regular, then by Lemma 1, R(C) ⊆ R(A) C

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implies that there exists X ∈ Fmm such that C = XA. Hence,     I I M = A = U A, where U = . X X For U − = [I 0], U − U = I and U − is a g-inverse of U . Thus U is regular. Now, M = U A is regular ⇒ (U A) M − (U A) = U A. Premultiplying with U − on both sides, we get A(M − U )A = A. Thus A is regular and (i) holds. (ii) can be proved in the same manner.  Combining Lemma 4 and Lemma 5 we get the following: Lemma 6. For A, B, C ∈ Fmn , the following statements hold:     A A 0 (i) If R(C) ⊆ R(A), then A is regular ⇔ is regular ⇔ C C 0 is regular.   A B (ii) If C(B) ⊆ C(A), then A is regular ⇔ [A B] is regular ⇔ 0 0 is regular.  A 0 with R(C) ⊆ R(A) and C D C(C) ⊆ C(D), block triangular g-inverse ⇐⇒  M is regular and M  has a lower  A 0 0 0 and M2 = Both M1 = are regular matrices. C 0 C D In this case, M = M1 + M2 and ρ(M ) = ρ(M1 ) + ρ(M2 ). Furthermore, M − = M1− + M2− is a g-inverse of M for suitable choice of ginverse of M1 and M2 . Theorem 3. For the matrix M =



Proof. M is a regular matrix and M has a lower block g-inverse ⇔ the blocks A and D are regular matrices (By Theorem 1), (By Lemma 6). ⇔ Both M1 and M2 are regular matrices ρ(M )

= ρ(A) + ρ(D) ( By Theorem 1 ), = ρ(M1 ) + ρ(M2 ).

By using C + C = C, we get, M = M1 + M2 . For the matrices M1 , M2 ,    −  0 A 0 0 − − and M2 = are g-inverses of M1 and M2 reM1 = ∗ 0 ∗ D− ,   0 A− − spectively, where “ ∗ ” is an arbitary matrix. Hence M1 = D− CA− 0

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0 0 are g-inverses of M1 and M2 respectively. D− CA− D− − − Furthermore, M1 + M2 = M − .

and M2− =



5. Fuzzy relational equations In this section, we discuss consistency of fuzzy matrix equation x · M = b,   A 0 where M = is a lower block triangular fuzzy matrix and x = [y z] C D and b = [c d] are partitions of x and b respectively in conformity with that of M.

Theorem 4. For the matrix M =



A C

0 D



such that R(C) ⊆ R(A), the

blocks A and D are regular matrices, the following statements are equivalent: (i) x · M = b is solvable. (ii) y · A = c, z · D = d are solvable and c ≥ dD− C.

Proof. (i) ⇒ (ii). Suppose x · M = b is solvable. Let α = [β γ] be a solution. The substitution gives β · A + γ · C = c and γ · D = d. Since R(C) ⊆ R(A), by using C = CA− A we get (β + γ CA− )A = c and γ · D = d. Therefore y · A = c and z · D = d are both solvable with the fact that y = β + γCA− is a solution of y · A = c and z = γ is a solution of z · D = d. Since D is regular, γ = dD− is a solution of z · D = d. Now, γC = dD− C. From βA + γC = c, by fuzzy addition we get c ≥ γC = dD− C as required. (ii) ⇒ (i). Suppose y · A = c and z · D = d are solvable. Since both A and D are regular, matrices y = cA− and z = dD− are the solutions of the equation y · A = c and z · D = d respectively. Hence, cA− A = c and dD− D = d. By fuzzy addition, c ≥ dD− C implies c + dD− C = c.   A 0 = [c A− A + dD− C dD− D] [c A− dD− ] C D = [c + dD− C = [c d] = b.

d]

Thus [c A− dD− ] is a solution of the equation x · M = b. Hence x · M = b is solvable.



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Remark 3. In Theorem 4, the conditions on M need not imply M is regular. Hence we have determined equivalent conditions for consistency of x · M = b without M being regular.

6. Triangular Toeplitz fuzzy matrices In this section, first we derive equivalent conditions for the idempotency of a triangular Toeplitz fuzzy matrix of orders upto 3. Then we discuss the idempotency of a general triangular Toeplitz fuzzy matrix of order k of the form 

a a·1 a·2 ·

   Tk =     ak−2 ak−1

0 a· a·1 · ak−3 ak−2

0 0· a· · · ak−3

· · · · · ·

· · · · · · · · · a · a1

0 0· 0· · 0 a



   .   

Theorem 5. (i) Each a ∈ [0, 1] is idempotent as well as regular.   a 0 is an idempotent matrix ⇔ a1 ≤ a. (ii) T2 = a1 a   a 0 0 (iii) T3 =  a1 a 0  is an idempotent matrix ⇔ a1 ≤ a2 ≤ a. a2 a1 a Proof. (i) is trivial. (ii) T2 is an idempotent matrix ⇔



a 0 a1 a a



=



a a1

0 a

⇔ a1 a = a1 ⇔ a1 ≤ a.   A 0 (iii) Let us partition T3 = , where C D   a 0 A= , D = [a], C = [a2 a1 a

a1 ].



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By (ii) the matrix A is idempotent ⇔ a1 ≤ a. D is idempotent.   a 0 = [a2 a1 ] CA = C ⇔ [a2 a1 ] a1 a ⇔ a2 a + a1 = a2 and a1 a = a1 ⇔ (a2 + a1 )a = a2 and a1 ≤ a DC = C ⇔ a[a2 a1 ] = [a2 a1 ] ⇔ a · a2 = a2 and a · a1 = a1 ⇔ a2 ≤ a and a1 ≤ a. Thus, CA = DC = C ⇔ a1 ≤ a, a2 ≤ a and (a1 + a2 ) · a = a2 ⇔ a1 ≤ a1 + a2 ≤ a, a2 ≤ a1 + a2 ≤ a and (a1 + a2 ) · a = a2 ⇔ a1 ≤ a, a2 ≤ a and a2 = (a1 + a2 ) · a = a1 + a2 ⇔ a1 ≤ a, a2 ≤ a and a1 ≤ a2 ⇔ a1 ≤ a2 ≤ a. By Corollary 3, T3 is an idempotent matrix ⇔ the blocks A, D are idempotent matrices and CA = DC = C ⇔ a1 ≤ a2 ≤ a.  Theorem 6. Tk is an idempotent matrix if a ≥ ak−1 ≥ ak−2 ≥ . . . ≥ a2 ≥ a1 . Proof. By Theorem 5, the matrices T1 , T2 and T3 are idempotent under the condition a ≥ a2 ≥ a1 . We prove that T4 is an idempotent matrix under the   A 0 condition a ≥ a3 ≥ a2 ≥ a1 . Let us partition T4 = , where A = T3 is C D an idempotent matrix. D = [a] is idempotent and C = [a3 a2 a1 ]. By Corollary 3, to prove that T4 is idempotent, it is enough to verify CA = DC = C.   a 0 0 CA = [a3 a2 a1 ]  a1 a 0  a2 a1 a = [(a3 · a) + (a1 · a2 ) (a2 · a) + a1 (a1 · a)] = [a3 a2 a1 ] since a ≥ a3 ≥ a2 ≥ a1 = C. Similarly, DC = C. Hence T4 is an idempotent matrix. By induction on k, we can prove that Tk is an idempotent matrix under the given condition. 

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Remark 4. If T4 is an idempotent matrix, then proceeding as in the proof of Theorem 5, we see that each a1 , a2 , a3 ≤ a, a1 ≤ a2 but a2 and a3 are not comparable. Hence we provide an example to show that the converse of Theorem 6 is not true.   1 0 0 0    ·3 1 0 0  A 0   Example 2. T4 =  = , where ·5 ·3 1 0  C D ·4 ·5 ·3 1   1 0 0 A =  ·3 1 0  , D = [1], C = [·4 · 5 · 3]. ·5 ·3 1 Since 1 ≥ ·5 ≥ ·3, by Theorem 5 A is an idempotent matrix and D = [1] is idempotent.   1 0 0 CA = [·4 · 5 · 3]  ·3 1 0  = [·4 · 5 · 3] = C, DC = 1 [·4 · 5 · 3] = C. ·5 ·3 1 Hence by Corollary 3, T4 is an idempotent matrix. However, the condition a2 ≤ a3 fails. References 1. H. H. Cho, Regular fuzzy matrices and fuzzy equations, fuzzy sets and systems 105 (1999), 445 - 451. 2. Cornelius T. Leondes, Fuzzy Theory Systems: Techniques and Applications, Vol. II. Academic Press, 1999. 3. K. H. Kim and F. W. Roush, Generalized Fuzzy Matrices, Fuzzy Sets and Systems 4 (1980), 293-315. 4. C. D. Meyer, Generalized Inverses of Block triangular Matrices, SIAM. J. Appl. Math. 19(4) (1970), 741-750. 5. Timothy J. Ross, Fuzzy Logic with Engineering Applications, McGraw Hill Inc. 1995. 6. Witold Pedrycz, Fuzzy Relational Calculus, Hand Book of Fuzzy Computations, IOP. Ltd.,1998. 7. C. K. Zhao, Inverses of L Fuzzy Matrices, Fuzzy Sets and Systems. 34 (1990), 103-116. AR. Meenakshi received her M. Math. from University of Waterloo (Canada) and Ph. D at Annamalai University. Since 1973, she has been at Annamalai University as Teaching Faculty, and she is the Dean, Faculty of Science from 2002. She was the receipient of Tamil Nadu State Council for Science and Technology, Scientist award for the year 1999. Her research interests focus on Generalized inverses of matrices, structure theory of Fuzzy Matrices and operators with applications. Department of Mathematics, Annamalai University, Annamalainagar-608002, India e-mail : arm [email protected]