SUMMARY
The purpose of this report is to analyze, from different points of view, two widely used classes of matrices, matrices with positive inverse and B-matrices. We will generalize, in some cases and in others complete the results obtained by different investigators.
Characterize the problem of inverse-positive matrices has been extensively discussed in the literature. Several authors studied the concept for inverse-positive matrices were also Z-matrix (i.e., M-matrices). Other authors took care to characterize the patterns of signs that have the inverse-positive matrices.
The inverse-positivity of real square matrices plays an important role in different areas of science and engineering and has been discussed in different contexts. In our work we present new characterizations of inverse-positive matrices. We also analyze the positive-inverse concept for a particular type of sign pattern: the checkerboard pattern.
The sub-direct sum of matrices is a generalization of the usual sum of matrices. It was introduced by C. Johnson and S. Fallat and appears naturally in matrix completion and overlapping subdomains in domain decomposition methods, among other contexts. Also appears in several variants of preconditioning additive Schwartz, and when analyzed by Schwartz additive methods for Markov chains. In this paper we provide new results on sub-direct sum of matrices with positive inverse and sub-direct sum of different classes of B-matrices, answering Fallat and Johnson questions for the classes of matrices mentioned.
Johnson studied the possible sign patterns consistent with a inverse-positive matrix. Following their findings, we discuss the concept of inverse-positive for a particular type of pattern: the checkerboard pattern. We study herein the inverse-positive status of lower (upper) triangular matrices with checkerboard pattern.
The Hadamard product of matrices has been investigated by several authors. Our purpose here is to study whether the Hadamard product of A and B and A with the inverse of B when A and B are inverse-positive matrices is also an inverse-positive. Regarding the patterns of signs, we will focus especially on the checkerboard pattern. Also studied Hadamard product for the B-matrices and its related classes (DB-matrix, B0-matrix and |B|-matrix).
In this manuscript we also provide some new results on the completion of partial B-matrices. We introduce a series of restrictions on the values of the entries in the matrices, of their signs or its associated graphs in order to close the problem in some directions.