Nearly L-matrices and generalized row sign balanced matrices Original Research Article

A real matrix A is called an L-matrix if every matrix with the same sign pattern as A has linearly independent columns. A nearly L-matrix is a matrix which is not an L-matrix, but each matrix obtained by deleting one of its columns is an L-matrix. A generalized row sign balanced (GRSB) matrix is a matrix which can be transformed to a matrix having both positive and negative entries in each row by multiplying some of its columns by −1. In this paper, we study the relations between L-matrices, nearly L-matrices and GRSB matrices. We obtain a complete characterization of nearly L-matrices in terms of GRSB matrices. By comparing this result with a well-known theorem about L-indecomposable, barely L-matrices, we find an interesting duality relation between L-matrices and GRSB matrices. We also use GRSB matrices to characterize the conditional S*-matrices (which are closely related to nearly L-matrices and the conditionally sign solvable linear systems). Finally, we propose some unsolved problems for further research.