THE BOOLEAN IDEMPOTENT MATRICES. Hong Youl Lee and Se Won Park. 1. Introduction

Transcription

1 J. Appl. Math. & Computing Vol. 15(2004), No pp THE BOOLEAN IDEMPOTENT MATRICES Hong Youl Lee and Se Won Park Abstract. In general, a matrix A is idempotent if A 2 = A. The idempotent matrices play an important role in the matrix theory and some properties of the Boolean matrices are examined. Using the upper diagonal completion process, we give the characterization of the Boolean idempotent matrices in modified Frobenius normal form. AMS Mathmatics Subject Classification : 15A15. Key words and phrases : Matrix, Boolean idempotent, Frobenious normal form. 1. Introduction In this paper we present the characterization of the Boolean idempotent matrices. For a pioneering work on the matrix theory, see [1],[4],[5],[6]. For further development of theory of Boolean matrices, the reader is referred to [2],[3]. In order to develop the properties of Boolean idempotent matrix, we must begin with the concept of Boolean algebra. The definition of a Boolean algebra which we are about to present is based on the structure introduced by E. V. Huntington in The Boolean algebra of two elements is most frequently used in applications and all other finite Boolean algebras are simply direct sums of copies of it. Moreover we show that by homomorphisms the theory of matrices over any Boolean algebra reduces to the two elements case. Thus we will primarily work with the two elements Boolean algebra in this paper. We shall use β 0 to denote the set {0, 1} with three opertations +,, c defined by as follows: Received September 4, Revised January 15, This paper was supported by Woosuk University. c 2004 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 475

2 476 Hong Youl Lee and Se Won Park 0+0=0 1=1 0=0 0=0, 1+0=0+1=1+1=1 1=1, 0 c =1, and 1 c =0. By a Boolean matrix we mean matrix over β 0. Such a matrix can be interpreted as a binary relation. Here we shall say what a Boolean matrix is and how Boolean matrices are added and multiplied. (The method is really the same as for matrices of complex numbers except that addition and multiplication of individual entries is Boolean). Definition 1.1. Let A be an n n Boolean matrix. The matrix A is Boolean idempotent if A 2 = A Since the order of the matrix is clear from the context, most of the time we suppress the order of the matrix. Matrix addition and multiplication are the same as in the case of complex matrices but the concerned sums and products of elements are Boolean. The concepts such as transpose, symmetricity, and idempotency, etc., are the same as in the case of real or complex matrices. Example 1.2. If A = , B = , then A + B = , AB = The matrix of the composition of two binary relations will be the Boolean product of the matrices of the relations. Furthermore, we can interpret a Boolean matrix as a graph. Definition 1.3. The adjacency matrix A G of a digraph G is (0, 1)-matrix such that a ij =1if there is an arc from vertex v i to v j and a ij =0otherwise. Dually, a digraph G is determined by the Boolean matrix A G. 2. Irreducible Boolean idempotent matrices

3 The Boolean Idempotent Matrices 477 First we examine some basic properties of Boolean idempotent matrices. They are useful in the following discussion. We know that all 1 1 Boolean matrices are idempotent. Hence, we deal only with a square Boolean matrix of order n 2. Let B I be the set of all Boolean idempotent matrices. In this paper, the set of numbers {1, 2, 3, n 1,n} will be denoted by N. Lemma 2.1. The set of all Boolean idempotent matrices, B I, is closed under the following operations: (i)permutation similarity; and (ii)transposition A matrix A of order n 2 is said to be reducible if there exists a permutation matrix P such that ( ) PAP T B C = O D where B and D are nonvacuous square matrices. Otherwise A is called irreducible. Remark 2.2. Let A =[a ij ] B I. Then a ij = 1 if and only if there exists a k N such that a ik =1=a kj. This statement follows from the fact that since A =[a ij ] is idempotent, we must have that a ij = k=n k=1 a ika kj for all i, j N. Another way of writing this is to observe that for a matrix A with an i-th row vector R i and a j-th column vector C j, A B I if and only if for all ir i = j J R j where J = {k e k R i }, e k is an row vector in the standard basis. Similar for C j. Lemma 2.3. Let A =[a ij ] be a Boolean idempotent matrix of order n 2. If a ij =0for some i and j in N, then each product a ik a kj =0for all k in N. Proof. It is an immediate consequence of the Boolean matrix product. A Boolean matrix A =[a ij ] of order n 2 is said to be graphical- transitive if a ik 0 and a kj 0 imply that a ij 0 for some i and j in N. Theorem 2.4. If an n n Boolean matrix A is idempotent, then A is graphicaltransitive. Proof. It is an immediate consequence of the idempotence and Lemma 2.3. Lemma 2.5. If A =[a ij ] is an n n irreducible Boolean idempotent matrix, then A is entrywise nonzero.

4 478 Hong Youl Lee and Se Won Park Proof. Suppose that A =[a ij ] is an n n irreducible Boolean idempotent matrix. For any indices i and j in N, the irreducibility of A implies that there is a path from i to j, say,a ik1 a k1k 2 a kmj 0, where each k h, h =1, 2,,m, is in N. By repeatedly using the graphical-transitivity of A, it follows that a ij 0. Since i and j are arbitrary indices in N, we conclude A is entrywise nonzero. Clearly the matrix J n of order n whose entries are all ones is an irreducible Boolean idempotent matrix. From Lemma 2.5, we have the following theorem. Theorem 2.6. Let A be an irreducible Boolean matrix of order n. Then A B I if and only if A = J n. 3. Reducible Boolean idempotent matrices Let A be an n n matrix. Then either A is irreducible or there exists a permutation matrix P such that A 11 P T AP =... 0 A kk in which A ii, i =1,,k is either irreducible of order n i or zero matrix where n n k = n. This is called the Frobenious normal form (Fnf) of A. IfA is a reducible Boolean idempotent matrix in Frobenious normal form, then it is clear that each irreducible diagonal block of A is a Boolean idempotent matrix. In the remainder of this paper, we assume that all reducible matrices are in Fnfs. Also we use the results of above section and assume that each nonzero irreducible diagonal block A ii of A is entrywise 1, that is, A ii = J ni. Lemma 3.1. Let A be an n n reducible Boolean idempotent matrix in Fnf, and let A ii and A jj be nonzeros. If n i n j matrix A ij contains a zero entry, then A ij is a 0-block. Proof. Assume A ij contains a zero entry (A ij ) kr = 0 for some k in N i and some r in N j. Then (A 2 ) kr = 0 and it follows that (A ii ) ks (A ij ) sr = 0 for some s in N i. However, (A ii ) ks 0 for all s in N i implies that (A ij ) sr = 0 for all s in N i. Consequently the r-th column of A ij is an entrywise 0-column. Similarly (A ij ) km (A jj ) mr = 0 for all m in N j. Since (A jj ) mr 0 for all m in N j, we conclude that (A ij ) km = 0 for all m in N j. Thus the k-th row of A ij is entrywise zero. Since (A ij ) kr is an arbitrary 0-entry in the r-th column of A ij, we conclude that every row of A ij is a 0-row, that is, A ij is a 0-block.

5 The Boolean Idempotent Matrices 479 Lemma 3.2. If A ii and A jj are entrywise nonzero Boolean matrices and A ij is an n i n j entrywise nonzero matrix, then (i) A ii A ij is defined if and only if each column A ij contains only 1 s and (ii) A ij A jj is defined if and only if each row A ij contains only 1 s. Proof. For simplicity, let A ii = H = [h ij ] and A ij = B = [b ij ]. Then (b 1j b 2j b nj ) T is the j-th column of B for all j in N j and (h 11 h 12 b 1ni ) is the first row of H. Consequently n i (HB) 1j = h ik b kj is defined if and only if b kj = h 1k for all k in N j. Thus (colj) B =(row1) T H.We omit the proof of (ii), since the argument used is similar to the one used to prove (i). k=1 Combining Lemma 3.1 and 3.2, we obtain the following lemma. Lemma 3.3. If A is an n n reducible Boolean matrix such that A ii and A jj are entrywise nonzero diagonal blocks, then A is Boolean idempotent only if the entries of A ij are obtained as follows: (i) A ij contains only 1 s or (ii) A ij is 0-block. Lemma 3.4. Let A be an n n reducible Boolean idempotent block matrix with an entrywise nonzero diagonal block A ii and a 0-block A jj. If A ij contains a 0-entry, then A ij contains a 0-column. Proof. To simplify notation, let A ii = H =[h ij ] and A ij = B =[b ij ]. Assume b rj = 0 for some r in N j. By Boolean idempotence, we know that the product entry (HB) rj = 0 and n i h rk b kj =(HB) rj =0. k=1 However, h rk 0 implies that b kj = 0 for all k in N i. Since a Boolean matrix is idempotent only if each off-diagonal block of A 2 is defined, we obtain the following: Lemma 3.5. Let A be an n n reducible Boolean idempotent matrix. If A ii is entrywise nonzero and A jj is an 0-block, then A is Boolean idempotent only if A ij =0or A ij contains only 1 s.

6 480 Hong Youl Lee and Se Won Park We should remark that if A ii is an 0-block and A jj is an entrywise nonzero matrix, then we can state results for A ij analogous to those given in Lemma 3.4 and 3.5. In the remainder of this paper, we refer to the latter results as Lemma 3.4 (ii) and 3.5(ii). Suppose A =[a ij ] is an n n reducible Boolean idempotent matrix in Frobenious normal form containing m diagonal blocks. Further assume A contains t consecutive diagonal blocks, where A ii (1 i<m) is the first of the t consecutive 0-diagonal blocks. It is not difficult to show that the t adjacent diagonal blocks are the diagonal entries of a t t 0-diagonal block in A. We relabel and denote this t t 0-block by A ii, and the diagonal 0-entries in A ii by A i1i 1,A i2i 2,,A iti t. Repeat this labeling procedure, if necessary, unit all consecutive diagonal blocks in A have been relabeled as descibed above. Call this the modified Frobenius normal form of A. We note that the diagonal blocks of a Boolean matrix in modified Frobenius normal form are entrywise nonzero or entrywise zero matrices. Since reducible Boolean matrix A in Frobenius normal form can be relabeled as described above, we may assume, without loss of generality, that A is in modified Frobenius normal form. If A is an m m block reducible matrix, then the off-diagonal blocks A i,i+k lie on the k-th superdiagonal for all k =1, 2,,m 1. Due to the triangular structure of A, each P i,i+k in the product matrix P = A 2 is independent of all terms above the k-th superdiagonal. This independence allows us to complete the zero pattern of A so that A = A 2, as described in the following: Algorithm 3.6 (The upper diagonal completion process). Let A =[A ij ] be an m m reducible and partial block Boolean idempotent matrix in modified Frobenius normal form. We can determine the entry of each off-diagonal block as follows: (i) Start with 1-st superdiagonal. Determine the entries of each off-diagonal block A i,i+1 using L- emma 3.3 if A ii and A i+1,i+1 are entrywise nonzero, Lemma 3.5 for each diagonal block of A i+1,i+1 if A ii is an entrywise nonzero and A i+1,i+1 is a 0-block, or Lemma 3.5 (ii) for each diagonal block of A ii so that A i 1,i A i,i+1 is unambiguously defined if A ii is a 0-block and A i+1,i+1 is an entrywise nonzero Boolean matrix. Move up to the next diagonal (if there is one). (ii) For each unspecified entry A i,i+k on the k-th superdiagonal, k = 2, 3,,m 1, if P i,i+k = A ii A i,i+k + A i,i+k A i+k,i+k use step (i) with i + k replacing it, otherwise let A i,i+k = A i,i+1 A i+1,i+k. When all blocks are specified on this diagonal, move up to the next diagonal, if there is one, increase k by 1 for all k =2, 3,, m 2, and repeat (ii).

7 The Boolean Idempotent Matrices 481 Example 3.7. Let A be a Boolean idempotent pattern as follow ; ?? A = ?? where is 0 or 1 and? is determined by the specified number that is on the lower superdiagonal than it. In the above pattern, let A be a Boolean idempotent matrix. Then using the algorithm 3.6, we can determine the entries and? in A to be a Boolean idempotent matrix A = Theorem 3.8. Let A be an n n reducible Boolean idempotent matrix in modified Frobenious normal form where each of whose diagonal blocks is entrywise nonzero or 0-block. Then A is Boolean idempotent only if each off-diagonal block A ij is obtained using Algorithm 3.6. Proof. The result follows from Lemma 3.3, 3.5, and 3.5(ii). To establish that completing each off-diagonal block of a reducible Boolean idempotent matrix A by the upper diagonal completion process is sufficient for Boolean idempotence, we use a graph-theoretic approch in the next section. 4. A Graph-theoretic Interpretation If A is a reducible Boolean idempotent matrix in modified Frobenius normal form and if A ii and A jj are entrywise 1, then according to the upper diagonal

8 482 Hong Youl Lee and Se Won Park completion process, A ij is a 1-matrix, J ij, or a 0-block. If A ii is a 0-block, each column of A ij is defined. If A jj is a 0-block, each row of A ij is defined. We now prepare to interpret the upper diagonal completion process graph-theoretically. Let A be a reducible Boolean matrix having m diagonal blocks. If each block entry in A is a 1-matrix or a 0-block, then we form the m m reduced matrix R =[r ij ]ofa as follows: { 1 if Aij is a 1-matrix, r ij = 0 if A ij is a 0-block for all i and j in N. The directed graph of the reduced matrix R is called the reduced directed graph of A, denoted by RD(A). In somewhat different terms, the reduced graph of a nonnegative matrix is defined in [4]. We say RD(A) is transitively closed if for any (i, j) and (j, k) in the edge set E, the edge (i, k) is in E. Lemma 4.1. Let A be a Boolean matrix in modified Frobenius normal form. If each off-diagonal block is obtained using the upper diagonal completion process, then RD(A) is transitively closed. Proof. Assume that A is a Boolean matrix that satisfies the conditions stated in the lemma. For contradiction, suppose that RD(A) is not transitively closed. Then there is a k such that i +1 k j i, where the edges (i, k), (k, j), (i, j) satisfy one of the two cases below: Case(i): Suppose the edges (i, k) and (k, j) are in the edge set E and (i, j) is not in E. Then A ij is a 0-matrix and A ik A kj is 1-matrix. By Algorithm 3.6 (ii), we have the following : A ij = A i,i+1 A i+1,j = A i,i+1 A i+1,i+2 A i+2,j = A i,i+2 A i+2,j. = A i,k A k,j = = A i,j 1 A j 1,j. However, A i,j = A i,k A k,j implies that A i,j 0 and (I,j) E. It is contradict to the assumption that (i, j) / E. Case(ii): Suppose the edges (i, j) E and either (i, k) E or (k, j) E, but not both. Assume (i, j) and (i, k) are in E, but (k, j) / E. Then A ij and A ik are 1-matrices and A kj is a 0-block. From case (i), we have A ij = A ik A kj 0 for all k = i +1,,j 1. Thus A ik 0 and A kj 0 and there are edges (i, k) and (k, j) ine which contradicts the assumption that (k, j) / E. A similar argument holds if (i, j) and (k, j) are in E, but (k, j) / E. Consequently, we conclude that RD(A) is transitively closed.

9 The Boolean Idempotent Matrices 483 Lemma 4.2. Let A be an n n reducible Boolean matrix in modified Frobenius normal form where each nonzero diagonal block is entrywise 1 and the entry of each off-diagonal block is determined using the upper diagonal completion process. Then A is idempotent. Proof. We use the result of Lemma 4.1 to assume, without loss of generality, that RD(A) is transitively closed. First assume A ij = 0 for any i and j in the index set N; and for contradiction, suppose there is a k such that i+1 k j 1 and A ik A kj 0. Then (i, k) and (k, j) are in the edge set E of the directed graph RD(A). Since RD(A) is transitively closed, (i, j) E, this implies A ij 0. However, this contradicts the assumption that A ij = 0. Thus P ij = A ii A ij A ij A jj = =0=A ij Now, suppose that A ij 0. Case(i): Assume that A ii and A jj are enrywise 1. Then A ii A ij and A ij A jj are 1-matrices and for each k such that i+1 k j 1, A ik A kj = A ij follows by the same argument as given in the proof of Lemma 4.1-case (i) and we conclude that P ij = A ij. Case(ii): Assume that A ii is entrywise 1 and A jj is a 0-block. Then A ii A ij is 1-matrix and A ik A kj = A ij for all k such that i +1 k j 1 as in case (i), so that P ij = A ij. Case(iii): Assume that A ii is a 0-block and A jj is entrywise 1. Then reversing the roles of A ii and A jj in case (ii) implies P ij = A ij. Case(iv): Assume that A ii and A jj are 0-blocks. By step (ii) of the completion process, we know A ij = A i,i+1 A i+1,j 0. Further, by argument given in case (i) of the proof, A ik A kj = A ij for all k such that i +1 k j 1andwe conclude P ij = A ij. Cases(i)-(iv) imply that P ij = A ij for any indices i and j in N, and it follows that A is idempotent. At the end of Section 3, in Theorem 3.8, we proved that it was necessary to determine the entry of each off-diagonal block A ij of a reducible Boolean idempotent matrix using the upper diagonal completion process. Lemma 4.2 implies that the completion process is sufficient for Boolean idempotence. Consequently we have the following: Theorem 4.3. A reducible Boolean matrix A in modified Frobenius normal form where each of whose nonzero diagonal block is entrywise 1 is idempotent if and only if the entry of each off-diagonal block is obtained using the upper diagonal completion process.

10 484 Hong Youl Lee and Se Won Park Example 4.4. The matrix A in Example 3.7 is a reducible Boolean idempotent matrix, since the entry of each off-diagonal block is determined using the completion process. References 1. L. B. Beasley, S. G. Lee, and S. W. Park, Weights of Idempotent Matrices, Trends in Mathematics 4(2001), K. H. Kim, The Number of Idempotents in (0, 1)-Matrix Semigroups, Liner Algebra and Its Appl. 5(1972), K. H. Kim, Boolean Matrix Theory and Application, Marcel Dekkr, Inc., S. G. Lee and S. W. Park, The Allowance of Idempotent of Sign Pattern Matrices, Comm. Korea Math. Soc. 10(1995), S. W. Park, L. B. Beasley, and S. G. Lee, Idempotence of (1, -1)-Matrices, Congressus Numerantium 146(2000), S. W. Park, S. G. Lee, and S. Z. Song, Nonnegativity of Reducible Sign Idempotent Matrices, Korea J. Comput. & Appl. Math. 7(2)(2000), Hong Youl Lee received his MS and Ph. D degree from SungKyunKwan University and have been at WooSuk University since His research interests focus on the operator theory and their application. Department of Mathematics Education, Woosuk University, Wanju-gun, Korea, hylee@core.woosuk.ac.kr Se Won Park received his MS and Ph. D degree from SungKyunKwan University and have been at SeoNam University since His research interests focus on the matrix theory and their application. Department of Mathematics, Seonam University, Namwon, Korea, swpark@seonam.ac.kr