Linear Algebra and its Applications 294 (1999) 85±92 www.elsevier.com/locate/laa Irreducible sign k-potent sign pattern matrices Je€rey Stuart a,* , Carolyn Eschenbach b,1 , Steve Kirkland c,2 a Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS 39406-5045, USA b Department of Mathematics and Computer Science, Georgia State University, Atlanta, GA 303033083, USA c Department of Mathematics and Statistics, University of Regina, Regina, Sask., Canada S4S 0A2 Received 11 December 1996; accepted 17 March 1999 Submitted by H. Schneider Abstract The sign pattern matrix A is called sign k-potent if k is the smallest positive integer for which Ak‡1 ˆ A: We characterize the irreducible pattern matrices that are sign k-potent and provide a canonical form for such matrices. Ó 1999 Published by Elsevier Science Inc. All rights reserved. Keywords: Sign k-potent; Sign pattern matrix, Reduced block matrix 0. Introduction If only the signs of the entries of a square, real matrix A are given, then generally the sign pattern of A2 will be unpredictable, although not entirely arbitrary. Relatively little is known about the possible sign correlations between the ambiguously signed entries of A2 ; or about the locations of the ambiguous entries. Consequently, the sign pattern problem for A2 was identi®ed in [5] as a model problem in qualitative matrix theory. Further, knowledge of the sign pattern of A2 ; of some higher power of A, or of some function of A, * Corresponding author. E-mail: je€.stuart@usm.edu E-mail: ceschenbach@cs.gsu.edu 2 E-mail: kirkland@math.uregina.ca 1 0024-3795/99/$ ± see front matter Ó 1999 Published by Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 - 3 7 9 5 ( 9 9 ) 0 0 0 7 6 - 2 86 J. Stuart et al. / Linear Algebra and its Applications 294 (1999) 85±92 would be useful in the intermediate stages of a number of qualitative matrix theory problems (see [4,7]). Note that the sign pattern problem for A2 is multifaceted, and that only a small portion of it was addressed in [5]. Speci®cally, the possible digraphs for the unambiguously signed entries of A2 , over all sign matrices A with entries in f‡; ÿg; were characterized. In this paper, we investigate the naturally related question of what sign pattern matrices A have unambiguous powers satisfying Ak‡1 ˆ A for minimal positive integer k. This paper is organized as follows: Section 1 introduces the basic ideas of patterns and their supports. Section 2 contains a pair of working lemmas. In section 3, sign k-potence is de®ned. Section 4 focuses on basic properties of sign k-potent matrices. It contains Lemma 5, which relates the sign k-potence of A and the values of p for which Ap‡1 ˆ A: This section also contains Theorem 7, which says that all powers of an irreducible, sign k-potent sign pattern matrix are unambiguous. Section 5 contains the main result of the paper, Theorem 8, which is a characterization theorem for all irreducible, sign k-potent matrices. Section 6 introduces the notion of a reduced block matrix. In Section 7, two special matrices, Pm and Qm ; are de®ned and are shown to be canonical forms for the reduced block matrix of an irreducible, sign k-potent matrix. 1. Patterns and their supports In this paper, we will be concerned with real matrices, sign pattern matrices and generalized sign pattern matrices. That is, our matrices will have entries respectively from R, from f‡; ÿ; 0g or from f‡; ÿ; 0; #g where # denotes an entry of ambiguous or unspeci®ed sign. For the multiplicative and additive rules governing the symbols ‡; ÿ; 0 and #, see [7]. In later sections, where it is contextually clear, we will use matrix or pattern matrix to mean sign pattern matrix. A pattern matrix A is called a subpattern matrix of a pattern matrix B if either A equals B or else A can be obtained from B by replacing one or more nonzero entries with a zero. For a matrix A (real, pattern or generalized pattern), the support of A, denoted by supp A†, is the set f i; j† : aij 6ˆ 0g: Note that for a pattern matrix A, supp A† is the support of every real matrix B in the pattern class of A. For a generalized sign pattern A, supp A† is the union of the supports of all real matrices B such that all of the following hold: sign bij † ˆ ÿ when aij ˆ ÿ; sign bij † ˆ ‡ when aij ˆ ‡; and sign bij † ˆ 0 when aij ˆ 0: For a matrix A (real, pattern or generalized pattern), de®ne j Aj to be the pattern matrix such that the i; j-entry of j Aj is ‡ whenever aij 6ˆ 0, and zero otherwise. Note that for any matrix A (real, pattern or generalized pattern), supp A† ˆ supp j Aj†: If A is a pattern matrix, and if B is a real matrix in the pattern class of A, then supp B† ˆ supp j Aj†: If A is a generalized pattern J. Stuart et al. / Linear Algebra and its Applications 294 (1999) 85±92 87 matrix, and if B is a real matrix in the pattern class of A, then supp B†  supp j Aj†: Observe that if A and B are real matrices of appropriate dimensions, then supp AB† ˆ supp j ABj†  supp j Ajj Bj†: Further, the only way that the containment can be proper is if at least one entry of AB is zero as a result of additive cancellation. Consequently, the following useful lemmas hold: Lemma 1. If A and B are pattern matrices or generalized pattern matrices of appropriate dimensions, then supp AB† ˆ supp j ABj† ˆ supp j Ajj Bj†: Further, if AB is itself a pattern matrix, then for all real matrices C in the pattern class of AB, supp C† ˆ supp j Ajj Bj†: Finally, if AB is a generalized pattern containing at least one #, then there exist real matrices C in the pattern class of AB such that supp(C) is a proper subset of supp j Ajj Bj†: Lemma 2. If A is a square pattern matrix and if k is a positive integer, then supp Ak † ˆ supp j Ajk †: Further, if Ak is itself a pattern matrix, then for all real matrices B in the pattern class of A, supp Bk † ˆ supp j Ajk †: 2. Patterns with a strictly nonzero row or column Suppose  that A and B are two patterns. Suppose that AB†ij is unambiguous. Let S ˆ k : aik bkj 6ˆ 0 : Then either for all k 2 S; aik ˆ bkj ; or else, for all k 2 S; aik ˆ ÿbkj : Suppose that the ith row of A; Ai has no zero entries. Then either column Bj has no zero entries, and hence, Bj ˆ ATi ; or else Bj has one or more zero entries, and it is obtained from ATi by zeroing entries. This discussion and its analog for rows yields the next lemma. Lemma 3. Let A and B be two patterns such that AB is unambiguous. If A has a row R that contains no zero entries, then B equals or is a subpattern of RT V for some sign row vector V with entries in f‡; ÿg: If B has a column C that contains no zero entries, then A equals or is a subpattern of WC T for some sign column vector W with entries in f‡; ÿg. Notice that the partial transitivity and the partial symmetry of these relationships place restrictions on the sign patterns of the remaining rows of A and on the remaining columns for B: These relationships are more fully explored in [5] for pattern matrices with no zero entries. 3. Sign k-potent patterns The square pattern matrix A is called sign k-potent if there exists a smallest positive integer k such that Ak‡1 ˆ A: (If k ˆ 1; then A is called sign 88 J. Stuart et al. / Linear Algebra and its Applications 294 (1999) 85±92 idempotent.) If A is an irreducible pattern matrix, then it has been shown in [3] that A is sign idempotent if and only if A is cyclically nonnegative and A is entrywise nonzero. Sign k-potence is a natural generalization of sign idempotence. Note that sign k-potence is closely related to notions of base and power investigated in [7]. The square sign pattern matrix A is called powerful if Am is unambiguous for every positive integer m. Sign k-potence motivated the study of powerful pattern matrices investigated in [7]. Example 4. The sign pattern matrices A and B given below are both sign 4potent and powerful. This is clear for B; and A can be obtained from B via signature similarity by S ˆ diag ÿ; ‡; ‡; ‡; ÿ; ‡† and permutation similarity by the permutation matrix corresponding to the permutation (1)(2)(5)(634). 2 3 0 ÿ 0 0 0 ÿ 60 0 ‡ 0 ÿ 07 6 7 60 0 0 ‡ 0 07 6 7; Aˆ6 0 0 0 07 6ÿ 0 7 40 0 0 ÿ 0 05 0 0 ‡ 0 ÿ 0 2 0 60 6 60 Bˆ6 60 6 40 ‡ ‡ 0 0 0 0 0 ‡ 0 0 0 0 0 0 ‡ ‡ 0 0 0 0 ‡ ‡ 0 0 0 3 0 07 7 07 7: ‡7 7 ‡5 0 Suppose that A is an irreducible pattern matrix. Suppose that A has index of imprimitivity m for some m P 2: It is well known (see [1, Section 2.2] or [2, Section 3.4], for example) that A is permutation similar to an m  m block partitioned, block-circulant pattern matrix of the form 2 3 0 ... 0 0 A1 0 6 0 0 ... 0 7 0 A2 6 .. 7 .. .. 6 7 6 0 . 7 . 0 . bˆ6 . 7; † A . . . 6 . 7 .. .. .. 0 7 6 . 6 7 .. 4 0 0 . 0 Amÿ1 5 Am 0 0 ... 0 0 b is unique up to permutation where the diagonal blocks are square. Further, A within the blocks and up to cyclic permutation of the sequence of the blocks. b given by () is called the cyclic form of A. When m ˆ 1, A is its The matrix A b ˆ A1 : own cyclic form, and it will be understood that A ˆ A J. Stuart et al. / Linear Algebra and its Applications 294 (1999) 85±92 89 Since permutation similarity preserves sign k-potence, it should be clear from circulant nature of () that there is a close relationship between the sign k-potence of A and the index of imprimitivity of A. 4. Results concerning sign k-potent matrices Lemma 5. Let A be an irreducible pattern matrix. Suppose that Ap‡1 ˆ A for some positive integer p. Then A is sign k-potent for some k, and k divides p: Proof. There are two cases: A is the 1  1 zero matrix , and A is nontrivial. Suppose that A is the 1  1 zero matrix. Clearly, Ap‡1 ˆ A for all positive integers p; and since sign p-potence uses the minimum choice of p; A is sign 1-potent. Clearly 1 divides p: Suppose that A is a nontrivial, irreducible matrix such that Ap‡1 ˆ A for some positive integer p. Then A is clearly sign k-potent for some positive integer k: By Theorem 3.3 of [7], A is powerful, and hence, by Lemma 1.2 of [7], k divides p:  Lemma 6. Let A be a sign k-potent pattern matrix. Then Atk‡1 ˆ A for all positive integers t. In particular, Atk‡1 is unambiguous for all positive integers t. Proof. Induct on t:  Lemma 6 coupled with Lemma 3.1 of [7] yields the following result that relates sign k-potent patterns and powerful patterns. Theorem 7. Let A be an irreducible pattern matrix. If A is sign k-potent, then A is powerful. 5. Characterization theorem for irreducible, sign k-potent patterns The main result in this section is a characterization of all irreducible k-potent pattern matrices. This result is related to Corollaries 2.4 and 3.6 of [7], which can be used to prove the result in the case that the index of imprimitivity, m, is odd. Theorem 8. Let A be an irreducible, sign k-potent pattern matrix with block b given by () with m P 1. For 1 6 i 6 m, let Ji denote the m  m cyclic form A matrix of pluses that has the same size as Ai ; and let Ci denote the first column of Ai : Then: m 2 fk=2; kg, and there exist vectors ui in f‡Ci ; ÿCi g for 1 6 i 6 m such that Ai ˆ ui uTi‡1 for 1 6 i 6 m ÿ 1 (when m P 2) and Am ˆ aum uT1 where a ˆ ‡ when m ˆ k, and a ˆ ÿ when m ˆ k=2. Further, if D ˆ diag u1 ; u2 ; . . . ; um †; then 90 J. Stuart et al. / Linear Algebra and its Applications 294 (1999) 85±92 2 b ÿ1 DAD 0 6 0 6 6 6 0 ˆ6 6 .. 6 . 6 4 0 aJm J1 0 .. . 0 J2 0 0 .. 0 0 0 0 . 0 0 .. . .. . .. .








0 .. . 0 0 0 0 .. . 3 7 7 7 7 7: 7 0 7 7 Jmÿ1 5 † 0 b ÿ1 ˆ aJ1 :) Conversely, given m P 1; given a set of (Note that when m ˆ 1; DAD strictly nonzero vectors ui for 1 6 i 6 m, and given a 2 f‡; ÿg; every matrix A whose block cyclic form has its blocks Ai specified by the above construction must be irreducible and sign k-potent, where k ˆ m when a ˆ ‡ and k ˆ 2m when a ˆ ÿ. Proof. Suppose that A is sign k-potent. By Theorem 7, each power of A is unambiguously de®ned. Since A is sign k-potent, Lemma 2 implies j Ajk‡1 ˆ j Aj; and thus jAj is sign m-potent for some m dividing k: Applying Corollary 2.4 of [7] to the irreducible pattern matrix j Aj, j Aj has block m  m cyclic form of type () such that each of the nonzero blocks is strictly positive. Thus each nonzero b for A is strictly nonzero. By Theorem 7, block in the block m  m cyclic form A b 2 are both unambiguous; consequently, each the pattern matrix A2 ; and hence, A of the products A1 A2 ; A2 A3 ; . . . ; Amÿ1 Am and Am A1 is unambiguous. Since each Ai is strictly nonzero, the results of Section 2 apply to each of the products. Speci®cally, let u2 ˆ C2 ; a strictly nonzero column of A2 : Since A1 A2 is unambiguous, applying Lemma 3, A1 ˆ u1 uT2 for some strictly nonzero vector u1 : In particular, since the ®rst column of A1 is strictly nonzero, u1 2 f‡C1 ; ÿC1 g: Since A1 A2 is unambiguous and A1 has a strictly nonzero row u2 †, it follows by Lemma 3 that every column of A2 is a multiple of u2 ; and hence, A2 ˆ u2 uT3 for some strictly nonzero vector u3 : Continuing this argument yields Ai ˆ ui uTi‡1 for 2 6 i 6 m ÿ 1; where ui 2 f‡Ci ; ÿCi g and Am ˆ um vT1 for some strictly nonzero vector um 2 f‡Cm ; ÿCm g and for somestrictly nonzero vector v1 . The product Am A1 ˆ um uTm v1 †uT1 is unambiguous, and hence, v1 is a multiple of um . Let v1 ˆ aum where a 2 f‡; ÿg: Then Am ˆ aum uT1 . It is straightforward to con®rm that m ˆ k when a is chosen to be positive, and that m ˆ k=2 when a is chosen to be negative. b ÿ1 are ‰diag ui †Š ui uT †‰diag ui‡1 †Š ˆ Ji Note that the nonzero blocks of DAD i‡1 ÿ1 for 1 6 i 6 m ÿ 1 since diag ui‡1 † ˆ diag ui‡1 † for each i. Also, ‰diag um †Š aum uT1 †‰diag u1 †Š ˆ aJm . Again, it is straightforward to check that the correct relationship between m and k holds. The converse follows from observing that the circulant block structure together with Ap‡1 ˆ A requires p to be a multiple of m. The construction of the nonzero blocks guarantees that k 2 fm; 2mg; with the particular value determined by the value of a:  J. Stuart et al. / Linear Algebra and its Applications 294 (1999) 85±92 91 6. Reduced block matrix Every square, generalized sign pattern matrix admits a symmetric block partition such that each block has the form aJ where a 2 f‡; ÿ; 0; #g and J is the all ‡ matrix of the appropriate size: simply employ all 1  1 blocks. We are interested in a coarsest block partition, that is, one that is not a proper subpartition of any other block partition. Lemma 9. Let A be a square, generalized sign pattern matrix. Then A has a unique coarsest symmetric block partition such that each block is of the form aJ where a 2 f‡; ÿ; 0; #g and J is the all ‡ matrix of the appropriate size. Proof. If A has two coarsest symmetric partitions, P1 and P2 then some block in the re®ned partition P1 [ P2 is assigned di€erent signs under partitions P1 and P2 ; a contradiction.  For a square, generalized sign pattern matrix A; the reduced block matrix for A; denoted red A†; is the matrix of signs induced by the coarsest partitioning of A: that is, if the blocks of A in a coarset partition are aij Jij for 1 6 i; j 6 m for some m; then red A† is the m  m matrix whose ij-entry is aij : Note that if red A† and A are both m  m; then A ˆ red A†: Theorem 10. Let A be a square generalized sign pattern matrix. Then for each k positive integer k; red Ak † ˆ red ‰red A†Š †: Further, if Ak‡1 ˆ A for some posik‡1 tive integer k, then ‰red A†Š ˆ red A†: Proof. The ®rst statement is an immediate consequence of the fact that the block partitioning is symmetric and that Jhi Jij ˆ Jhj for all h; i and j with k‡1 1 6 h; i; j 6 m. If Ak‡1 ˆ A; then red A† ˆ red Ak‡1 † ˆ red ‰red A†Š †: But k‡1 k‡1 since red ‰red A†Š † and ‰red A†Š are the same size, they must be equal.  k‡1 It is unclear whether ‰red A†Š ˆ red A† always implies that Ak‡1 ˆ A since for n > 2; it is easy to construct a pair of n  n sign pattern matrices A and B with red A† ˆ red B† but with A 6ˆ B: Consequently, it is unclear whether sign k-potence of a generalized sign pattern matrix A implies sign k-potence of red A† due to the minimality requirement on k. Nonetheless, it will be shown in the next section that in the case of irreducible sign pattern matrices, Theorem 10 usually leads to a reduction in the order of matrices when testing for sign k-potence. 7. Pn and Qn Let Pn denote the n  n circulant permutation pattern matrix with ‡'s on the ®rst superdiagonal and ‡ in the n; 1- position. Let Qn denote the pattern matrix 92 J. Stuart et al. / Linear Algebra and its Applications 294 (1999) 85±92 obtained from Pn by replacing the ‡ in the n; 1-position with a ÿ: Apparently, P1 ˆ ‰‡Š and Q1 ˆ ‰ÿŠ, the signed 1  1 pattern matrices. Note that since Pn and Qn are signed permutations, they are invertible as sign patterns. Further, each of the matrices Pn and Qn is its own unique reduced block matrix. Finally, note that Pn is sign n-potent and that Qn is sign 2n-potent. Theorem 8 can be reinterpreted as: Theorem 11. Let A be an irreducible sign pattern matrix. Then A is sign k-potent if and only if A can be transformed via signature similarity and permutation similarity into a pattern matrix B such that red B† is either Pm or Qm for some m: If k is odd, then m ˆ k and red B† ˆ Pk . If k is even, then either m ˆ k and red B† ˆ Pk , or else m ˆ k=2 and red B† ˆ Qk=2 . Thus the set of matrices Qn and Pn for positive integers n provides a set of canonical forms for the irreducible, sign k-potent matrices. This will be exploited in a subsequent paper devoted to the structure of reducible, sign k-potent sign pattern matrices, [6]. References [1] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. [2] R.A. Brualdi, H.J. Ryser, Combinatorial Matrix Theory, Cambridge University Press, New York, 1991. [3] C. Eschenbach, Idempotence for sign pattern matrices, Linear Algebra Appl. 180 (1993) 153± 165. [4] C. Eschenbach, F. Hall, C. Johnson, Self-inverse sign patterns, IMA Vol. Math. Appl. 190 (1993) 169±179. [5] C. Eschenbach, F. Hall, C. Johnson, Z. Li, The graphs of the unambiguous entries in the product of two (‡; ÿ) sign pattern matrices, Linear Algebra Appl. 260 (1997) 95±112. [6] J. Stuart, Reducible sign k-potent sign pattern matrices, submitted to Linear Algebra Appl. [7] Z. Li, F. Hall, C. Eschenbach, On the period and base of a sign pattern matrix, Linear Algebra Appl. 212/213 (1994) 101±120.