Inverse-Positive Matrices with Checkerboard Pattern

Http Dx Doi Org 10 1080 03081087 2010 547495, 2011

For a square (0, 1, −1) sign pattern matrix S, denote the qualitative class of S by Q(S). In this paper, we investigate the relationship between sign patterns and matrices that diagonalise an irreducible nonnegative matrix. We explicitly describe the sign patterns S such that every matrix in Q(S) diagonalises some irreducible nonnegative matrix. Further, we characterise the sign patterns S such that some member of Q(S) diagonalises an irreducible nonnegative matrix. Finally, we provide necessary and sufficient conditions for a multiset of real numbers to be realised as the spectrum of an irreducible nonnegative matrix M that is diagonalised by a matrix in the qualitative class of some S 2 N S sign pattern.

Linear Algebra and its Applications, 2001

An m-by-n matrix A is called totally nonnegative if every minor of A is nonnegative. The Hadamard product of two matrices is simply their entry-wise product. This paper introduces the subclass of totally nonnegative matrices whose Hadamard product with any totally nonnegative matrix is again totally nonnegative. Many properties concerning this class are discussed including: a complete characterization for min{m, n} < 4; a characterization of the zero-nonzero patterns for which all totally nonnegative matrices lie in this class; and connections to Oppenheim's inequality.

Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, 2011

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.

Applied Mathematics and Computation, 2011

A nonsingular real matrix A is said to be inverse-positive if all the elements of its inverse are nonnegative. This class of matrices contains the M-matrices, from which inherit some of their properties and applications, especially in Economy and in the description of iterative methods for solving nonlinear systems. In this paper we present some new characterizations for inverse-positive matrices and we analyze when this concept is preserved by the sub-direct sum of matrices.

Linear Algebra and its Applications, 2008

Generalizations of M-matrices are studied, including the new class of GM-matrices. The matrices studied are of the form sI − B with B having the Perron-Frobenius property, but not necessarily being nonnegative. Results for these classes of matrices are shown, which are analogous to those known for M-matrices. Also, various splittings of a GM-matrix are studied along with conditions for their convergence.

Linear Algebra and its Applications, 1979

The concepts of matrix monotonicity, generalized inverse-positivity and splittings are investigated and are used to characterize the class of all M-matrices A, extending the well-known property that A-* > 0 whenever A is nonsingular. These conditions are grouped into classes in order to identify those that are equivalent for arbitrary real matrices A. It is shown how the nonnegativity of a generalized left inverse of A plays a fundamental role in such characterizations, thereby extending recent work by one of the authors, by Meyer and Stadehnaier and by Rothblnm. In addition, new characterizations are provided for the class of M-matrices with "property c"; that is, matrices A having a representation A = sI-I?, s > 0, B > 0, where the powers of (l/s)B converge. Applications of these results to the study of iterative methods for solving arbitrary systems of linear equations are given elsewhere. 1. INTRODUCTION AND PRELIMINARIES All vectors and matrices considered in this paper are real. For notation purposes let R" denote the vector space of all n-tuples of real numbers and R n,n denote the set of all n x n matrices of real numbers. If x E R n and each component of x is nonnegative, we shall write x > 0, and if each component of A E R"," is nonnegative we shall write A > 0, where the symbol 0 denotes the zero vector or zero matrix. Of particular interest here will be the set Zn~"={A=(aii)ER"~":a,j~OforaUi#j}.

Electronic Journal of Linear Algebra, 2009

An M ∨ -matrix has the form A = sI -B, where s ≥ ρ(B) ≥ 0 and B is eventually nonnegative; i.e., B k is entrywise nonnegative for all sufficiently large integers k. A theory of M ∨ -matrices is developed here that parallels the theory of M-matrices, in particular as it regards exponential nonnegativity, spectral properties, semipositivity, monotonicity, inverse nonnegativity and diagonal dominance.

Linear Algebra and its Applications, 1997

Linear Algebra and its Applications, 2007

Considered are continuous, positive Hadamard powers of entry-wise positive (nonnegative) matrices. Those that are eventually (in the sense of all Hadamard powers beyond some point) totally positive, totally nonnegative, doubly nonnegative and doubly positive are characterized. For example, for matrices with at least four rows and four columns, Hadamard powers greater than one of totally positive matrices need not be totally positive, but they are eventually totally positive.

2020

It is shown that for positive real numbers 0 &lt; λ1 &lt; · · · &lt; λn, [