Powers of Ray Pattern Matrices

We examine the similarities and differences between results for powers of sign pattern matrices and for powers of ray pattern matrices. In particular, we investigate what is known about k-potent patterns and powerful patterns in both the irreducible and reducible cases. 1 Signs, Rays, Sign & Ray Patterns Since at least the 1960’s, the qualitative theory of real matrices has been an extremely fruitful area of research. When posing qualitative questions, we ask what aspects of a matrix — such as stability, invertibility, controllability — are determined entirely by the sign pattern — the arrangement of positive, negative and zero entries — without reference to the magnitudes of the entries. During the last decade, there has been a vigorous effort to develop a corresponding qualitative theory of complex matrices. One of the main approaches has been to recognize that positivity and negativity naturally generalize to rays of fixed argument in the complex plane. That is, extending the view that all positive numbers as equivalent to the number +1, and all negative numbers as equivalent to the number −1, we will treat the complex ray consisting of all complex numbers of the form re with r > 0 and θ ∈ R as equivalent to e. Of course, we need to introduce an appropriate arithmetic on such rays. Clearly, r1e iθ + r2e iθ will be equivalent to e provided r1 > 0 and r2 > 0. In general, however, the choice of θ for r1e iθ1 + r2e iθ2 to be equivalent to e will depend not only on θ1 and θ2, but also on r1 and r2. Consequently, we will require an additional symbol, #, to represent the result of adding nonzero complex numbers that lie on distinct rays. A ray operation that results in # will be called ambiguous. The complete arithmetic properties of # can be found ∗Department of Mathematics, Pacific Lutheran University, Tacoma, WA 98447 USA, [email protected] in [11] or [6]. If A is a fixed, m × n real matrix, the qualitative class of A, called the sign pattern of A, is the set of all m×n real matrices B such that sign(bij) = sign(aij) for all i and j. For convenience, the class is often represented by an appropriate canonical element: the unique member of the class whose entries are in {0,−1,+1}. An m×n generalized sign pattern is obtained by replacing one or more entries in a canonical representative of a sign pattern by the symbol #. The real matrix B is an element of a generalized sign pattern exactly when sign(bij) agrees with the the sign of each corresponding unambiguous entry of the canonical representative. If A is a fixed, m × n complex matrix, the qualitative class of A, called the ray pattern of A, is the set of all m × n complex matrices B with the same zerononzero pattern as A such that arg(bij) = arg(aij) (modulo 2π) whenever aij = 0. For convenience, the class is often represented by the unique member of the class whose entries are in S = {z ∈ C : |z| ∈ {0, 1}}. An m × n generalized ray pattern is obtained by replacing one or more entries in a canonical representative of a ray pattern by the symbol #. That is, by allowing one or more entries in the matrices of the pattern to have unspecified argument (or be zero). A pattern B is called a subpattern of a pattern A if either B equals A or else B can be obtained from A by replacing one or more nonzero entries with a zero. Two special patterns will be used extensively: In, the n × n identity, and J , the m× n matrix all of whose entries are 1’s. When working with pattern classes, it is often useful to know what transformations send sign (ray) patterns to sign (ray) patterns. Clearly, scalar multiplication and transposition do so. While matrix multiplication generally does not transform one sign (ray) pattern into another, multiplication by certain patterns does. In particular, preor post-multiplication by a pattern whose canonical representative is a permutation matrix either fixes a pattern, or transforms it into another pattern. Thus permutation equivalence and permutation similarity are useful transformations. Diagonal scaling, that is, preor post-multiplication by a nonsingular diagonal pattern, sends patterns to patterns. Observe that nonsingular, diagonal sign patterns are self-inverses as patterns; and that nonsingular, diagonal ray patterns are invertible as patterns via conjugation. Thus another useful pattern transformation is diagonal similarity. 2 Some Qualitative Questions Among the many questions concerning sign and ray patterns, we will briefly mention one and then focus on another. Perhaps the oldest question about sign patterns is which sign patterns are invertible? Work on the invertibility of sign patterns extends back to work by Bassett, Maybee and Quirk ([1]), and includes much other work including [4] and [8]. During the last ten years, attention has shifted to the subject of ray patterns and invertibility, see [7], for example. The question on which this paper focuses, is what can be said about powers of patterns? Suppose that A is a square matrix, real or complex, certainly A is well-defined. However, if A is a sign or ray pattern, what exactly does A mean? If A is a square sign or ray pattern, we will denote by A the (possibly generalized) sign or ray pattern corresponding to the set of all products of two real or complex matrices from the pattern A. One natural place to begin is, for what patterns is A = A, or more generally, A = A for some positive integer k? Such patterns are called sign k-potent sign patterns, or pattern k-potent ray patterns. Sign k-potent patterns were investigated by Eschenbach, Hall, Johnson and Li [3] in the strictly nonzero case with k = 2; by Stuart, Eschenbach and Kirkland [12] in the irreducible case for general k, and by Stuart [9] in the reducible case. The corresponding results for pattern k-potent ray patterns was investigated by Stuart, Beasley and Shader [11] in the irreducible case; and by Stuart [10] in the reducible case. The irreducible patterns have been fully characterized. Necessary conditions (partial characterizations) are known for the reducible patterns. It is easy to show that if A = A for some positive integer k where A is either a sign pattern or a ray pattern, then in fact A is actually a sign pattern or ray pattern for every positive integer h. Thus naturally, we are lead to the question: For what sign (ray) patterns is A actually a sign (ray) pattern for every positive integer h? Such patterns are called powerful. Powerful sign patterns were investigated by Li, Hall and Eschenbach in [5]. They showed that if A is an n × n powerful pattern, then A is periodic, that is, there existed unique, smallest positive integers b and p, called the base and period, respectively, such that A = A, and that p < 3. Notice that when b = 1, A is sign p-potent. Also, notice that if the sign pattern A has base b and period p, then A = A. Letting M = A, it follows that M = M, so that M is sign k-potent for some k that divides p. A complete characterization for the irreducible, powerful sign patterns is given in [5], The case of powerful ray patterns is more complicated than that of powerful sign patterns. The simple 1x1 example A = [e] is clearly powerful but there exist no positive integers b and p such that A = A. Li, Hall and Stuart showed in [6] that for the irreducible, powerful ray pattern A, there always exists a θ ∈ R such that B = eA has base and period. Returning to the preceding example, θ = −1, and B = [1]; clearly B has base b = 1 and period p = 1. If the powerful ray pattern A has base b and period p, then A = A. Letting M = A, it follows that M = M, so that M is pattern k-potent for some k that divides p. Consequently, we would expect that the structure of pattern k-potent ray patterns is closely related to the structure of powerful ray patterns. For convenience, we note here that the set of powerful ray patterns is closed under scalar multiplication by a ray, diagonal similarity, permutation similarity, transposition, conjugation, direct sums and the formation of subpatterns. 3 Canonical Forms for Irreducible Pattern k-Potent Ray Patterns Suppose that A is a square, irreducible ray pattern. It is well known (see [2, Section 3.4], for example) that there is a unique largest, positive integer m, called the index of imprimitivity of A, such that A is permutation similar to an m × m block partitioned, ray pattern of the form:  =   0 A1 0 0 . . . 0 0 0 A2 0 . . . 0 0 . . . 0 . . . .. .. . . . . . . . . . 0 0 0 . . . 0 Am−1 Am 0 0 . . . 0 0   . (*) where the diagonal blocks are square. Further,  is unique up to permutation within the blocks and up to cyclic permutation of the sequence of the blocks. The matrix  given by (*) is called the cyclic form of A. When m = 1, A is its own cyclic form, and it will be understood that A =  = A1. It should be apparent that if the irreducible, ray pattern A is in cyclic form with index of imprimitivity m, and that if A is pattern k-potent, then m divides k. Let Pn denote the n×n circulant permutation matrix with first row (0, 1, 0, . . . , 0). Let w be a primitive h root of unity for some positive integer h. (By convention, w = 1 is a primitive 1 root of unity.) Then wPn is invertible as a ray pattern, with inverse ray pattern (wPn) ∗ = wP n . Also note that n is the smallest positive integer such that P n = In, implying that (wPn) s = In whenever s is divisible by both h and n. In particular, = lcm(h, n) is the smallest such positive integer. Thus, wPn is pattern -potent. Every square, generalized ray pattern admits a symmetric block partition such that each block has the form αJ where α ∈ S ∪ {#} and J is the all ones matrix of the appropriate size. A coarsest block partition of this type is one that is not a proper subpartition of any other symmetric block partition with blocks of the form αJ. It was noted in [10] that the coarsest block partition of this type always exists and is unique for generalized ray patterns. For a square, generalized ray pattern A, the reduced block matrix for A, denoted red(A), is the unique ray pattern induced by the coarsest partitioning of A. That is, if the blocks of A in a coarsest partition are αhjJhj for 1 ≤ h, j ≤ m for some m, then red(A) is the m × m ray pattern whose (h, j)-entry is αhj . Observe that wPn for w ∈ S is its own unique, reduced block matrix. The following results are Theorem 1 and Theorem 2 of [10]. Theorem 1. Let A be a square, generalized ray pattern. Then for each positive integer k, red(A) = red([red(A)]). Further, if A = A for some positive integer k, then [red(A)] = red(A). Theorem 2. Let A be a nontrivial, irreducible ray pattern. Let k be a positive integer. Then A is pattern k-potent if and only if A can be transformed via signature similarity and permutation similarity into a ray pattern B such that red(B) is wPm where m is some positive integer that divides k, and where w is a complex number such that w is a primitive (k/m) root of unity. Note that diagonal similarity preserves cyclic form. Thus the previous result can be restated as follows: Theorem 3. Let A be a nontrivial, irreducible ray pattern with index of imprimitivity m. Suppose that A is in cyclic form (*). Then A is pattern k-potent if and only if, after an appropriate diagonal similarity, Ah = wJh for 1 ≤ j ≤ m, where Jh has the same dimensions as Ah, and where w m is a primitive (k/m) root of unity. 4 Canonical Forms for Irreducible Powerful Ray Patterns Theorem 3.4 and the remarks preceding Theorem 3.5 of [6] yield: Theorem 4. Let A be a nontrivial, irreducible ray pattern with index of imprimitivity m. Suppose that A is in cyclic form (*). Then A is powerful if and only if, after an appropriate diagonal similarity, there exists a ray w such that Ah is a subpattern of wJh for 1 ≤ j ≤ m, where Jh has the same dimensions as Ah. Notice that if A is powerful with index of primitivity m, and if w is the ray given in the preceding theorem, then w−1A is either pattern m-potent, or else the blocks w−1Ah can be filled in to yield a pattern m-potent ray pattern. Consequently, the following result, which is Theorem 3.6 of [6], is not surprising. Apparently raising B to the b power as given by the condition B = B is to fill in the zeros in the blocks Bh. Theorem 5. Let A be an irreducible ray pattern. Then A is powerful if and only if there is a ray c such that B = cA is periodic. We close this section with a surprising consequence of the preceding results. Corollary 6. An irreducible, powerful ray pattern is always a subpattern of an entrywise nonzero, powerful ray pattern. 5 Cyclic Normal Form for Pattern k-Potent Ray Patterns The rest of this paper discusses the powers of reducible ray patterns. Suppose that A is permutation similar to [ 0m×m 0m×n 0n×m B ]