Linear operators that strongly preserve commuting pairs of fuzzy matrices

Fuzzy Sets and Systems 41 (1991) 167-173 North-Holland

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Linear operators that strongly preserve commuting pairs of fuzzy matrices* LeRoy B. Beasley Utah State University, Logan, UT 84322-3900, USA

Norman J. Pullman Queen's University, Kingston, Ontario, Canada K7L 3N6 Received April 1989 Revised October 1989

Abstract: For each n I> 3, we characterize the linear operators, T, on the n x n fuzzy matrices that preserve commutativity strongly (that is, T(X) commutes with T(Y) if and only if X commutes with Y, for all X and Y). Characterizations of the corresponding linear operators on the fuzzy symmetric matrices are also obtained. Keywords: Fuzzy matrix; linear operator; commuting matrices; preserver.

1. Introduction and summary T h e set of c o m m u t i n g pairs of matrices, (~, is the set of ( u n o r d e r e d ) pairs of matrices (X, Y) such that X Y = Y X . T h e linear o p e r a t o r T is said to preserve (S (or simply T preserves commutativity), w h e n e v e r T ( X ) T ( Y ) = T ( Y ) T ( X ) if X Y = Y X ; T strongly preserves ~ w h e n T ( X ) T ( Y ) = T ( Y ) T ( X ) if and only if X Y = YXo In 1976 Watkins [6] p r o v e d that if n / > 4, d~ is the set of n x n matrices over an algebraically closed field of characteristic 0, and L is a nonsingular linear o p e r a t o r on ~ which preserves c o m m u t a t i v i t y , then there exists an invertible S in d~, a n o n z e r o scalar c, and a linear functional f such that either L ( X ) = cSXS-~ + f ( X ) I for all X • d/tn,,(~), or L ( X ) --- c S X t S -1 + f ( X ) l for all X in d~. In 1978, Beasley [1] e x t e n d e d this to the case n = 3. Also in [1] Beasley s h o w e d that the same characterization holds if n / > 3 and L strongly preserves commutativity. T h e real symmetric and c o m p l e x H e r m i t i a n cases were first investigated by C h a n and Lim [3] in 1982 w h e r e the s a m e results were established as in the general case with the exception that the invertible matrix, S, must be o r t h o g o n a l or unitary. F u r t h e r extensions and generalizations to m o r e general fields were o b t a i n e d by R a d j a v i [5] and Choi, Jafarian and R a d j a v i [4]. T h r o u g h o u t this article we are c o n c e r n e d with matrices with fuzzy scalar * This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant A4041. 0165-0114/91/$03.50 © 1991--Elsevier Science Publishers B.V. (North-Holland)

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entries. We shall use ' + ' to denote max and '.' or juxtaposition to denote min. With these conventions the standard definitions of linear operator, matrix multiplication, matrix addition, etc. apply. Evidently, the following operations strongly preserve the set of commuting pairs of fuzzy matrices: (a) transposition (X ~ X t); (b) similarity (X---~ S X S -1 for some fixed invertible matrix S); For each n ~>3 we show that the semigroup of linear operators strongly preserving commuting pairs of fuzzy matrices is generated by transposition and similarity. For each n 1> 3, we show that the semigroup of linear operators strongly preserving commuting pairs of symmetric fuzzy matrices consists of the similarity operators. The case n = 2 appears to be as intractable for fuzzy matrices as it is for matrices over fields.

2. The general case

In this section E denotes the set of n x n fuzzy matrices and Eq the member of E having all entries 0 except the (i, j)-th which is 1. We call Eq a cell. Denote the matrix in E whose entries are all 1 by J and its column vectors by j. The Schur (or Hadamard) product, A o B, of A, B in E is the matrix [aijbq]. Lemma 2.1 [2, Lemma 3.5]. If T strongly preserves the set of all (0, 1)-matrices in E which commute with J then there are permutation matrices, P and Q, such that either (i) T(X) = P X Q for all X ~ E or (ii) T(X) = p X t Q for all X ~ E . We shall denote by x E the set of all matrices in E with entries either 0 or x, i.e., x E = {A e E : aq ~ {0, x}}. We let ~x denote the set of all matrices in x E which commute with xJ. In the following lemmas, given a linear operator T, let mq denote the smallest nonzero entry in T(Eij), and m = min{mq: 1 <~i, j ~
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Lemma 2.3. Suppose T is a linear operator preserving commuting pairs of matrices in ~ . If m T is bijective on ~¢,, then T is one of or a composition o f (a) transposition, (b) similarity, or (c) scaling by T(J), i.e., X---~T(J) o X. Proof. Let wq denote the smallest nonzero entry in T(Eq), W = [wq] and m = min(wq: 1 ~< i, j ~< n}. Then m :~ 0. Applying L e m m a 2.2 to the operator (mT) acting on m E , we see that there is a permutation matrix P such that (i) (mT)(Eij) = e ( m E q ) e t, for all (i, j), or (ii) (mT)(Eq) = P(mEj~)P t, for all (i, j). In either case, (mT)(Eq) has only one nonzero entry, and hence so does T(Eq). Therefore T ( E q ) = P(wqEq)P t or P(wijEji)Pt according as (i) or (ii) holds. The lemma follows by the linearity of T because W = T(J). [] Suppose D is a diagonal matrix, A = D + m J, m :/: 0, B has no zero entries, B 2 = mJ and

M=

mJ

where A or B may be vacuous. Any matrix similar to M (i.e., of the form Q M Q t, Q a permutation matrix) will be called a commutable matrix. Lemma 2.4. If C is a commutable matrix then scaling by C (i.e., X---> C ° X ) preserves commuting pairs. Proof. A straightforward computation. Lemma 2.5. Suppose T is a linear operator preserving commuting pairs in ~ and n >! 3. If m T is bijective on m~t, then T(J) is commutable. Proof. Let W = T(J), m = min{wq: 1 ~< i, j <~ n} and m ' = min{wq: i :/:j}. Then m ' ~> m. We will first show that m ' = m. We observe that: (i) W commutes with T ( Z ) whenever J commutes with Z. Let j, l be distinct indices and P be the permutation matrix with Pii = Pjl = Po = 1 for all i :/:j, l. Then P commutes with J. Let G(l) = W ( W o T(P)). Then WilWlj

__ --

( l ) __ - wiiwij

gq

(2.1)

by (i). Then for all i, wi~ >- wijwii ~ m'. Therefore m = m'

(2.2)

and hence some off-diagonal entry in W takes on the value m. Next we prove the following observation. (ii) If wii > m, then every off-diagonal entry in row i and every off-diagonal entry in column i is m.

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W e k n o w that w~j = m for s o m e j 4=/. A p p l y i n g (2.1) we have m = w e w o = w o. So some entry in row i is m. N o w c h o o s e k 4= i and let A be the n x n matrix with % = a ~ , =ajk = a u = 1 for all 14=j; all o t h e r entries in A are 0. Let H = W ( W o A ) . T h e n applying (i) we have w/,wo = hjj = Wj, Wk~

(2.3)

w j i w i i = h j i = WjkWki.

(2.4)

and If w# > m, then wjiwii > m , so (2.4) implies (2.5)

w/, Wki > m .

We also have , ( k ) : WiiWik wowj* -__ 5~*

(2.6)

WkiW~i = g ~ ) = WkjWji.

(2.7)

and SO (2.7) and (2.5) imply (2.8)

WkjW/i > m .

But (2.3) implies WjkWkj = m since w0 = m and W/k > m by (2.5), contradicting (2.8). Thus wji = m. By (2.7), wki = m. B y (2.6), Wik = m . T h e r e f o r e W~k = Wki = m for all k ~ i. Next we prove the following observation. (iii) I f wii = m a n d w o > m f o r s o m e j, t h e n e v e r y entry in r o w j a n d c o l u m n i ism.

Let C be the matrix with ca = c# = cki = C/k = 1 for all l ~ k; all o t h e r entries in C are 0. Let F = C + E where all entries in E are 0 except e 0- = 1. T h e n C and F c o m m u t e with J. Letting S = ( C o W ) W and U = ( F o W ) W we obtain

WijWjj = Sij=

(2.9)

WiiWij,

Wi]W]k = Sik = WiiWik,

(2.10)

w # w i / = u//= w//,

(2.11)

WkiWi/ = Uk/ = Wk/W#,

(2.12)

But w~i = m , so (2.9) implies w H = m , (2.10) implies Wjk = m . Then (2.11) implies w/i = m and (2.12) implies Wk~ = m . Hence every entry in row j and column i is m. N o w we can simultaneously p e r m u t e the rows and columns of W so that the resulting matrix Q W Q t has the f o r m

where .4 is p x p. Also w h e n i ~ p then % = aj~ = m for all j 4= i, and for all i there is some j 4= i such that b 0 > m or bj~ > m. W e have shown in (iii) that when b 0 > m, then bjk = m = bki for all k. Let Ai = {j: b o > m } , then unless Ai = 0,

(B2),. = ~ bob/,.

(2.13)

J~/h

and hence (B2)ik = m for all i and k. T h a t is, B 2 = m J .

[]

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It follows from observation (iii) in the proof of the previous lemma, that when B 2 = mJ and B has no zero entries, then B is similar to a matrix of the form

where the diagonal blocks are square. This characterizes such matrices B. We call operators of the form X---) C o X commutable scaling operators when C is a commutable matrix. The previous lemmas then imply the following lemma. Lemma 2.6. f f n I> 3 then the semigroup of those linear operators T on preserving commuting pairs, such that m T is bijective on mJR is generated by transposition, similarity and commutable scaling. Note that the condition " m T is bijective on m E " replaces the statement that T is nonsingular (invertible) in the field case studied in [1] and [6]. Theorem 2.1. f f n/> 3 then the semigroup of linear operators on the n x n fuzzy matrices strongly preserving commuting pairs is generated by transposition and similarity. Proof. It follows from L e m m a 2.1 that m T is bijective. By L e m m a 2.6 and the observations in Section 1 we only need show that X - - - ) X o j is the only commutable scaling which strongly preserves commuting pairs in ~t. Let X ~ X o C be a commutable scaling with Cpq < 1. Let A be the matrix in ~ with a~k = aqp = Cpq for all k =/:p, q, apq = 1 and all other entries of A are zero. Let P be the matrix Cpqm. Now, A does not commute with J and P does, but A o C = P o C. Thus X ~ X o C does not strongly preserve commuting pairs. []

3. The symmetric case In this section we will investigate those linear operators on the set of fuzzy symmetric matrices that strongly preserve commuting pairs. Let 30 denote the set of all symmetric matrices in ~n.n- We define a digon-matrix to be the sum of a cell and its transpose. Notice that diagonal digon-matrices are cells. A k-star-matrix is the sum of k digon-matrices whose nonzero entries lie in row and column i, for some fixed i. Clearly all digon-matrices and all star-matrices are symmetric. Lemma 3.1 [2, L e m m a 4.3]. If T strongly preserves the set of all (0, 1)-matrices in d~ which commute with J then T preserves the set of all (0, 1)-star-matrices. As in the general case, we shall denote by x30 the set of all matrices in 3 0 with entries either 0 or x, i.e., xS = {A • S: aij • {0, x}}. We let ~x denote the subset of all matrices in x30 which commute with xJ. In the following lemmas, given a linear operator T, let mij denote the smallest nonzero entry in T(Eij), and m = min{mij: 1 <~ i, j ~< n}. Then m ~e 0.

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Lemma 3.2. Suppose T is a linear operator preserving commuting pairs o f matrices in 5e. I f m T is bijective on m 5 ° then m T is a similarity operator.

Proof. Since mT is bijective on m 5 ~, m T ( m J ) = mJ. Since m5¢ and ~m are finite, and since mT is bijective and preserves commuting pairs (because T does), mT(Cem) = ~m and mT((mSe)\~m)= (mSe)\~m. That is, m T strongly preserves ~m. Applying Lemma 3.1 to the operator m T (here, m plays the r61e of 1) we see that m T preserves (0, m)-star-matrices. Let o be the map of {1 . . . . , n} to itself defined by o(i) = j if and only if the n-star-matrix on row and column i is mapped to one on row and column j. Since m T is bijective, o is one-to-one, and hence onto. Let P be the permutation matrix corresponding to o. Then mT(X) = P X P t, for all X. [] The proof of the next lemma is obtained mutatis mutandi from the proof of Lemma 2.3, making the appropriate substitutions for the symmetric case, i.e. substituting 'E~j+Ej~' for 'Eq', '5¢' for ' ~ ' , etc., noting that for symmetric matrices X - ~ X t is the identity operator. Lemma 3.3. Suppose T is a linear operator preserving commuting pairs o f matrices in 6P. I f m T is bijective on 4,, then T is one o f or a composition o f (a) similarity, or (b) scaling by W, i.e., X ~ W o X, where W = T(J).

Suppose D is a block diagonal matrix whose diagonal blocks are either 1 x 1 matrices or 2 × 2 symmetric matrices with zero diagonal, and M = D + m J, m :/:0. Any matrix similar to M (i.e., of the form Q M Q t, Q a permutation matrix) will be called a symmetric-commutable matrix. Lemma 3.4. I f C is a symmetric-commutable matrix then scaling by C (i.e., X---~ C o X ) preserves commuting pairs in 5~.

Proof. A straightforward computation. Lemma 3.5. Suppose T is a linear operator preserving commuting pairs in 5¢ and n >13. I f m T is bijective on m ~ , then T(J) is symmetric-commutable. Proof. Let W = T ( J ) , m = m i n { w q : l < ~ i , j < ~ n } and m ' = m i n { w q : i : / : j } . Then m' t> m. We have m' = m by the arguments in the first few lines of Lemma 2.5. The following observations also follow by the same argument as in L e m m a 2.5. (i) W commutes with T(Z) whenever J commutes with Z. (ii) I f w, > m, then every off-diagonal entry in row i and every off-diagonal entry in column i is m. Next we prove the following observation. (iii) I f wii = m and wq = wji > m for some j :/: i, then every other entry in rows i and j and columns i and j is m.

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Since the f o r m u l a (2.1) was arrived at by c h o o s i n g a s y m m e t r i c matrix, for any l :/: i, j, we have witwtj -- wiiwij = m. H e n c e , (iii) holds since I was arbitrary. N o w we can simultaneously p e r m u t e the rows and c o l u m n s o f W so that the resulting matrix Q W Q t has the f o r m D + m J w h e r e D is a block-diagonal matrix whose diagonal blocks are either 1 x 1 matrices or 2 x 2 s y m m e t r i c matrices with zero diagonal. [] W e call o p e r a t o r s o f the f o r m X--> C o X symmetric-commutable scaling operators w h e n C is a s y m m e t r i c - c o m m u t a b l e matrix. T h e previous l e m m a s then imply the following lemma.

Lemma 3.6. / f n / > 3 then the semigroup of linear operators T on be preserving commuting pairs such that m T is bi]ective on mbe is generated by similarity and symmetric-commutable scaling. T h e p r o o f of the following t h e o r e m is the s a m e a r g u m e n t , mutatis mutandi, as that o f T h e o r e m 2.1.

Theorem 3.1. f f n I> 3 then the only linear operators on the symmetric n x n f u z z y matrices strongly preserving commuting pairs are similarity operators.

References [1] L.B. Beasley, Linear transformations on matrices: the invariance of commuting pairs of matrices, Linear and Multilinear Algebra 6 (1978) 179-183. [2] L.B. Beasley and N.J. Pullman, Linear operators that strongly preserve commuting pairs of Boolean matrices, Linear Algebra Appl. 132 (1990) 137-143. [3] G.H. Chan and M.H. Lim, Linear transformations on symmetric matrices that preserve commutativity, Linear Algebra Appl. 47 (1982) 11-22. [4] M.D. Choi, A.A. Jafarian and H. Radjavi, Linear maps preserving commutativity, Linear Algebra Appl. 87 (1987) 227-241. [5] H. Radjavi, Commutativity-preserving operators on symmetric matrices, Linear Algebra Appl. 61 (1984) 219. [6] W. Watkins, Linear maps that preserve commuting pairs of matrices, Linear Algebra Appl. 14 (1976) 29-35.