Characterizations of sign patterns of inverse-positive matrices

A result of Johnson, Leighton, and Robinson characterizing sign patterns of real matrices with nonzero entries whose inverses are (entrywise) positive is generalized. The restriction to matrices with nonzero entries is removed, and additional five equivalent conditions are established. One of them, using a graph-theoretical approach, expedites a simple criterion for recognition of such sign patterns.     

Required nonzero patterns for nonsingular sign regular matrices

An n×n real matrix is called sign regular if, for each k(1?k?n), all its minors of order k have the same nonstrict sign. The zero entries which can appear in a nonsingular sign regular matrix depend on its signature because the signature can imply that certain entries are necessarily nonzero. The patterns for the required nonzero entries of nonsingular sign regular matrices are analyzed.     

Construction and nonnegativity of sign idempotent sign pattern matrices

In this paper, we modify Eschenbach’s algorithm for constructing sign idempotent sign pattern matrices so that it correctly constructs all of them. We find distinct classes of sign idempotent sign pattern matrices that are signature similar to an entrywise nonnegative sign pattern matrix. Additionally, if for a sign idempotent sign pattern matrix A there exists a signature matrix S such that SAS is nonnegative, we prove such S is unique up to multiplication by -1 if the signed digraph D(A) is not disconnected.     

Complexity measures of sign matrices

In this paper we consider four previously known parameters of sign matrices from a complexity-theoretic perspective. The main technical contributions are tight (or nearly tight) inequalities that we establish among these parameters. Several new open problems are raised as well. Supported by the ISF. Supported by the ARC.     

Ranks of zero patterns and sign patterns

Let F be a field with at least three elements. Zero patterns P such that all matrices over F with pattern P have the same rank are characterized. Similar results are proven for sign patterns. These results are applied to answering two open questions on conditions for formal nonsingularity of a pattern P, as well as to proving a sufficient condition on P such that all matrices over F with pattern P have the same height characteristic.     

Alternating sign matrices and descending plane partitions

An alternating sign matrix is a square matrix such that (i) all entries are 1, ?1, or 0, (ii) every row and column has sum 1, and (iii) in every row and column the nonzero entries alternate in sign. Striking numerical evidence of a connection between these matrices and the descending plane partitions introduced by Andrews (Invent. Math.53 (1979), 193–225) have been discovered, but attempts to prove the existence of such a connection have been unsuccessful. This evidence, however, did suggest a method of proving the Andrews conjecture on descending plane partitions, which in turn suggested a method of proving the Macdonald conjecture on cyclically symmetric plane partitions (Invent. Math.66 (1982), 73–87). In this paper is a discussion of alternating sign matrices and descending plane partitions, and several conjectures and theorems about them are presented.     

Multiply-refined enumeration of alternating sign matrices

Four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices (ASMs). Specifically, these statistics are the positions of the 1’s in the first and last rows and columns of an ASM, and the numbers of generalized inversions and −1’s in an ASM. Previously-known and related results for the exact enumeration of ASMs with prescribed values of some of these statistics are discussed in detail. A quadratic relation which recursively determines the generating function associated with all six statistics is then obtained. This relation also leads to various new identities satisfied by generating functions associated with fewer than six of the statistics. The derivation of the relation involves combining the Desnanot–Jacobi determinant identity with the Izergin–Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions.     

Nonsingular almost strictly sign regular matrices

The class of nonsingular almost strictly totally positive matrices has been characterized [M. Gasca, J.M. Peña, Characterizations and decompositions of almost strictly positive matrices, SIAM J. Matrix Anal. Appl. 28 (2006) 1–8]. In this paper, we discuss the class of almost strictly sign regular matrices that includes almost strictly totally positive matrices. A characterization is provided for these matrices in terms of their nontrivial minors using consecutive rows and consecutive columns. In particular, we present a characterization of certain almost strictly sign regular matrices in terms of a very reduced number of boundary almost trivial minors.     

Eigenvalue perturbation theory of structured real matrices and their sign characteristics under generic structured rank-one perturbations

An eigenvalue perturbation theory under rank-one perturbations is developed for classes of real matrices that are symmetric with respect to a non-degenerate bilinear form, or Hamiltonian with respect to a non-degenerate skew-symmetric form. In contrast to the case of complex matrices, the sign characteristic is a crucial feature of matrices in these classes. The behaviour of the sign characteristic under generic rank-one perturbations is analyzed in each of these two classes of matrices. Partial results are presented, but some questions remain open. Applications include boundedness and robust boundedness for solutions of structured systems of linear differential equations with respect to general perturbations as well as with respect to structured rank perturbations of the coefficients.     

More refined enumerations of alternating sign matrices

We study a further refinement of the standard refined enumeration of alternating sign matrices (ASMs) according to their first two rows instead of just the first row, and more general “d-refined” enumerations of ASMs according to the first d rows. For the doubly-refined case of d=2, we derive a system of linear equations satisfied by the doubly-refined enumeration numbers A_n,i,j that enumerate such matrices. We give a conjectural explicit formula for A_n,i,j and formulate several other conjectures about the sufficiency of the linear equations to determine the A_n,i,j's and about an extension of the linear equations to the general d-refined enumerations.     

三幂等符号模式矩阵的结构

Abstract. A matrix whose entries are , -, and 0 is called a sign pattern matrix. For a signpattern matrix A,if A3 =A, then A is said to be sign tripotent. In this paper, the characteriza-tion of the n by n(n≥2) sign pattern matrices A which are sign tripotent has been given out.Furthermore, the necessary and sufficient condition of A3=A but A2≠A is obtained, too.     

Depth of almost strictly sign regular matrices

The concept of depth of an almost strictly sign regular matrix is introduced and used to simplify some algorithmic characterizations of these matrices.     

Quantum Knizhnik-Zamolodchikov equation, totally symmetric self-complementary plane partitions, and alternating sign matrices

We present multiple-residue integral formulas for partial sums in the basis of link patterns of the polynomial solution of the level-1 quantum Knizhnik-Zamolodchikov equation at arbitrary values of the quantum parameter q. These formulas allow rewriting and generalizing a recent conjecture of Di Francesco connecting these sums to generating polynomials for weighted totally symmetric self-complementary plane partitions. We reduce the corresponding conjectures to a single integral identity, yet to be proved. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 387–408, March, 2008.     

± sign pattern matrices that allow orthogonality

A sign pattern A is a ± sign pattern if A has no zero entries. A allows orthogonality if there exists a real orthogonal matrix B whose sign pattern equals A. Some sufficient conditions are given for a sign pattern matrix to allow orthogonality, and a complete characterization is given for ± sign patterns with n − 1 ⩽ N₋(A) ⩽ n + 1 to allow orthogonality.