Nearly L-matrices and generalized row sign balanced matrices (Q1587890)

A real matrix \(A\) is called an L-matrix if every matrix with the same sign pattern as \(A\) has linearly independent columns. \(A\) is a nearly L-matrix if it is not an L-matrix but each matrix obtained from \(A\) 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 \(-1.\) The authors give a thorough study of the relationship between L-matrices, nearly L-matrices, and GRSB matrices. Particularly, they obtain a complete characterization of nearly L-matrices in terms of GRSB matrices. Moreover, this class of matrices is used to characterize so-called conditional \(S^*\)-matrices which are closely related to nearly L-matrices and conditionally sign solvable linear systems. Three interesting open problems are proposed.