Carolyn A. Eschenbach

On the period and base of a sign pattern matrix

Abstract A square sign pattern matrix A (whose entries are +, −, or 0) is said to be powerful if all the powers A 1 , A 2 , A 3 ,…, are unambiguously defined. For a powerful pattern A , if A l = A l + p with l and p minimal, then l is called the base of A and p is called the period of A . We characterize irreducible powerful sign pattern matrices and investigate the period and base of a powerful sign pattern matrix. We also consider some connections with real matrices and give some significant classes of powerful patterns.

Sign patterns that require real, nonreal or pure imaginary eigenvalues

We characterize the n-by-n sign pattern matrices that require all real, all nonreal, and all pure imaginary eigenvalues. Characterization of sign patterns that allow a real eigenvalue and those that allow a nonreal eigenvalue then follow. Some related specialized results and a characterization of sign patterns that allow a positive real eigenvalue are included.

Potentially nilpotent sign pattern matrices

Abstract A sign pattern is said to be potentially nilpotent if it allows nilpotence. In this paper, a number of qualitative necessary or sufficient conditions for a sign pattern to allow nilpotence are established. The sign patterns that allow nilpotence of index 2 are investigated. For orders up to three, potentially nilpotent sign patterns are characterized. Potentially nilpotent tree sign patterns are also explored.

A combinatorial converse to the Perron-Frobenius theorem

Abstract A sign pattern requires (allows) the Perron property if every (some) matrix with that sign pattern has its special radius among its eigenvalues. We characterize those sign patterns that require the Perron property, which, in a sense, provides a converse to the classical Perron-Frobenius theorem. Also, a large class of patterns that allow the Perron property is identified, but a complete characterization remains an open problem.

Sign patterns that require repeated eigenvalues

Abstract Motivated by the question of which sign patterns allow a diagonalizable matrix, we relate a number of properties to that of requiring repeated eigenvalues. These are then used to make several observations about sign patterns that allow diagonalizability. The only barrier to diagonalizability is a required nontrivial Jordan structure associated with zero. We note that the question of sign patterns that require diagonalizability is also open.

From real to complex sign pattern matrices

This paper extends some fundamental concepts of qualitative matrix analysis from sign pattern classes of real matrices to sign pattern classes of complex matrices. A complex sign pattern and its corresponding sign pattern class are defined in such a way that they generalize the definitions of a (real) sign pattern and its corresponding sign pattern class. A survey of several qualitative results on complex sign patterns is presented. In particular, sign nonsingular complex patterns are investigated. The type of region in the complex plane representing the distribution of the determinants of the matrices in the sign pattern class of a sign nonsingular complex pattern is identified. Cyclically nonnegative complex patterns and complex patterns that are signature similar to nonnegative patterns are characterized. Extensions of sign stable and sign semistable patterns from the real to the complex case are given. Results on ray patterns are also obtained. Finally, many open questions are mentioned.

On Almost Regular Tournament Matrices

Abstract Spectral and determinantal properties of a special class M n of 2n×2n almost regular tournament matrices are studied. In particular, the maximum Perron value of the matrices in this class is determined and shown to be achieved by the Brualdi–Li matrix, which has been conjectured to have the largest Perron value among all tournament matrices of even order. We also establish some determinantal inequalities for matrices in M n and describe the structure of their associated walk spaces.

Idempotence for sign-pattern matrices

Abstract We first characterize the n -by- n irreducible sign-pattern matrices A that are sign idempotent, that is A = A 2 . To identify the n -by- n reducible sign idempotent patterns, we develop the upper diagonal completion process to find the sign pattern of each off-diagonal block in an upper block triangular sign-pattern matrix A , so that A = A 2 . Next we analyze the completion process qualitatively, and then discuss a graph-theoretic interpretation of it. We then formulate the first characterization of sign idempotent patterns in terms of the upper diagonal completion process. Finally we establish a graph-theoretic characterization of sign idempotent patterns.

Self-inverse sign patterns

We first show that if A is a self-inverse sign pattern matrix, then every principal submatrix of A that is not combinatorially singular (does not require singularity) is also a self-inverse sign pattern. Next we characterize the class of all n-by-n irreducible self-inverse sign pattern matrices. We then discuss reducible self-inverse patterns, assumed to be in Frobenius normal form, in which each irreducible diagonal block is a self-inverse sign pattern matrix. Finally we present an implicit form for determining the sign patterns of the off-diagonal blocks (unspecified block matrices) so that a reducible matrix is self-inverse.

Properties of Tournaments Among Well-Matched Players

1. TOURNAMENTS. In an n-player round robin tournament, each player plays one match against each of the other n - 1 players. The win-loss outcomes of these matches can be conveniently recorded in a tournament matrix A = [aij] as follows: First label the players in any order as 1, 2,.. ., n. For each pair i and j, set ai1 = 1 if player i defeats player j, and set aij = 0 otherwise. If i j j, then exactly one of aij and aji is nonzero; when i = j, aii = 0. What properties of the matrix A are related to the strengths of the players? The simplest measure of strength is the number of matches that the player wins, and the row sums of A count the number of matches won by each player. We are interested in understanding tournaments among players who are well matched in the sense that each player wins about half of the matches played. If the number of players is odd, many properties of A are very well understood. If the number of players is even, however, the properties of A are far less well understood. Indeed, there are many easily stated questions that lead to hard, open problems. Some of these problems are the focus of this paper. We let I denote the identity matrix, we let J denote the square matrix all of whose entries are ones, and we let e denote the column vector whose entries are all ones. A matrix A whose entries are zeros and ones is a tournament matrix exactly when