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Alternating Sign Matrices, Related (0,1)-Matrices, and the Smith

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Alternating Sign Matrices, Related (0; 1)-Matrices, and the Smith Normal Form Richard A. Brualdi∗ Geir Dahly March 11, 2018 Dedicated with admiration to Vladimir Sergeichuk on the occasion of his 70th birthday. Abstract We investigate the Smith Normal Form (SNF) of alternating sign matrices (ASMs) and related matrices of 0's and 1's ((0; 1)-matrices). We identify certain classes of ASMs and (0; 1)-matrices whose SNFs are (0; 1)-matrices. We relate some of our work to various ranks, in particular, the (0; 1)-rank of a (0; 1)-matrix, that is, the bipartite partition number of a bipartite graph. Key words. Alternating sign matrix, convex (0; 1)-matrix, rank, bipartite graph, Smith normal form. AMS subject classifications. 05B20, 15B48. 1 Introduction An alternating sign matrix, abbreviated to ASM, is an n × n (0; ±1)-matrix such that, ignoring 0's, the 1's and −1's in each row and column alternate beginning ∗Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. [email protected] yDepartment of Mathematics, University of Oslo, Norway. [email protected]. Correspond- ing author. 1 and ending with a 1. ASMs can be regarded as generalizations of permutation matrices, since permutation matrices are the ASMs without any −1's. Some ASMs A containing −1's are signature equivalent to (0; 1)-matrices, that is, there may exist diagonal matrices D1 and D2 all of whose diagonal entries are 1 or −1, such that D1AD2 is a (0; 1)-matrix. The matrix D1AD2 results from A by multiplying certain rows and columns by −1. The following example illustrates these observations. Example 1.1. 1 −1 1 −1 2 1 3 −1 1 6 1 1 1 7 1 1 −1 1 ! 6 7 : 4 1 1 1 5 −1 1 −1 1 1 1 1 Recall the three basic unimodular row operations on an integral matrix: (i) interchanging two rows, (ii) multiplying a row by −1, and (iii) adding an integral multiple of one row to another row. Operations (i), (ii) and (iii) are called elementary row operations, abbreviated to EROs. Unimodular column operations are defined similarly and we get ECOs. For simplicity, we refer to both of these operations as elementary operations or EOs. Thus D1AD2 results from A by a sequence of EOs of type (ii). Let A be an m × n integral matrix. Then, by EOs, A can be brought to diagonal form 2 3 e1 0 ··· 0 6 0 e2 ··· 0 Or;n−r 7 6 7 6 . .. 7 6 . 7 6 7 (1) 6 0 0 ··· er 7 6 7 4 5 Om−r;r Om−r;n−r where r is the rank of A, and e1; e2; : : : ; er are positive integers satisfying the divis- ibility conditions e1je2; e2je3; : : : ; er−1jer. The matrix (1) is called the Smith normal form of A, denoted as SNF(A), and e1; e2; : : : ; er are the invariant factors of A. The invariant factors of A are unique and, in fact, ei = di=di−1 where di is the greatest common divisor of all determinantal minors of size i (1 ≤ i ≤ r) and d0 = 1. The positive integers d1; d2; : : : ; dr are the elementary divisors of A. If A is nonsingular, then r = n and e1e2 ··· en = j det(A)j. Note that the SNF of a matrix is, in partic- ular, invariant under permutations of its rows and of its columns. The paper [18] surveys some aspects of the Smith normal form of combinatorially defined matrices. 2 We say that A has a (0; 1)-SNF provided that its invariant factors all equal 1, that is, provided I O SNF(A) = r (2) OO where r is the rank of A. Neither the Smith normal form of an ASM nor of a (0; 1)-matrix needs to be a (0; 1)-matrix as shown in the following examples. Example 1.2. We have 02 1 1 0 31 2 1 0 0 3 SNF @4 0 1 1 5A = 4 0 1 0 5 : (3) 1 0 1 0 0 2 This follows by observing that this matrix has determinant equal to 2 and all 2 × 2 minors have value ±1. Now consider the following 12 × 12 ASM A where we use + to denote a 1 and − to denote a −1: 2 + 3 6 + 7 6 7 6 + 7 6 7 6 + − + 7 6 7 6 + − + 7 6 7 6 7 6 + − + 7 A = 6 7 : 6 + − + 7 6 7 6 + − + 7 6 7 6 + − + 7 6 7 6 + 7 6 7 4 + 5 + The shaded entries of A indicate a cycle configuration of consecutive entries of the form (+ − +) such that any two have at most one entry in common. To find the SNF of A we use each of the 1's in the first three rows and eliminate the nonzeros below, and then we use each of the 1's in the first three columns and eliminate the 3 nonzeros to the right. The resulting matrix is 2 + 3 6 + 7 6 7 6 + 7 6 7 6 + + 7 6 7 6 + 7 6 7 6 7 6 + + 7 B = 6 7 : 6 + 7 6 7 6 + 7 6 7 6 + + 7 6 7 6 7 6 7 4 5 Note the 6 ones in shaded cells now form a bipartite cycle. Permuting rows and columns so that this cycle corresponds to rows and columns 7; 8; 9 we get 2 1 3 6 1 7 6 7 6 1 7 6 7 6 1 7 6 7 6 1 7 6 7 6 7 0 6 1 7 B = 6 7 : 6 1 1 7 6 7 6 1 1 7 6 7 6 1 1 7 6 7 6 7 6 7 4 5 Then (see (3)) SNF(B0) = SNF(A) is the diagonal matrix with diagonal (1; 1; 1; 1; 1; 1; 1; 1; 2; 0; 0; 0); and hence A does not have a (0; 1)-SNF. If an n×n integral matrix A satisfies that, for k = 1; 2; : : : ; n, A contains a minor which is ±1, then it follows that every elementary divisor, and hence every invariant factor, is 1, so that A has a (0; 1)-SNF. ASMs satisfying this sufficient condition may have some interesting properties. 4 Our goal in this paper is to initiate an investigation of the Smith Normal Form of ASMs and related (0; 1)-matrices. Note that a (0; ±1)-matrix A is signature equivalent to a (0; 1)-matrix if and only if there exist permutation matrices P and Q such that A −A P AQ = 11 12 (4) −A21 A22 where A11;A12;A21;A22 are all (0; 1)-matrices. The matrix A22, and hence A12 and A21, may be empty. With a (0; ±1)-matrix A we can associate a bipartite graph with vertex bipartition U (corresponding to the rows) and W (corresponding to the columns), and edges signed as +1 or −1, to get a signed bipartite graph G±(A). The matrix A is then the signed adjacency matrix of G±(A). A cycle of G±(A) is a positive cycle provided the product of the signs of its edges is +1, that is, has an even number of edges with sign −1, and is a negative cycle otherwise. It follows easily that if A is signature equivalent to a (0; 1)-matrix (see (4)), then all cycles of G±(A) are positive. The following known theorem [9, 13] asserts that the converse holds as well. Theorem 1.3. Let A be an m × n (0; ±1)-matrix. Then A is signature equivalent to a (0; 1)-matrix if and only if all cycles of G±(A) are positive. Proof. For the convenience of the reader we outline a brief proof of the converse. So suppose that all cycles of G±(A) are positive, that is, contain an even number of minus edges. Let G+ be the bipartite graph obtained from G± by deleting its minus edges. Then there does not exist a negative edge of G joining two vertices in the same connected component of G+, since such an edge would give a cycle of G with exactly one minus edge. Hence all the minus edges of G, if any, join two vertices in different components of G+. Let H be the graph whose vertices correspond to the connected components of G+ with an edge between two connected components if there is a negative edge between them in G±. Then H is a bipartite graph for, otherwise, H contains a cycle of odd length and then G± contains a negative cycle. The bipartition of the vertices of H induces the partition as given in (4). 2 Convex (0; 1)-Matrices and ASM's Let A = [aij] be a nonzero (0; ±1)-matrix. We say that A is row-convex if there does not exist a zero between two nonzeros in any row. The notion of column-convex is defined similarly. If A is both row- and column-convex, we say that A is convex. A (0; 1)-matrix which is either row-convex or column-convex is usually called an interval matrix. Row-convexity or column-convexity of a (0; 1)-matrix implies a (0; 1)-SNF. 5 Theorem 2.1. Let A be an m × n (0; 1)-matrix which is row-convex or column- convex. Then A has a (0; 1)-SNF. Proof. Assume that A is column-convex. Then A is an interval matrix, and this implies that A is a totally unimodular matrix [14].
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