Decomposition Theorems for Conditional Sign-Solvability and Sign-Solvability of General Systems

For each m × n real matrix A, let $\Null(A)$ be the right nullspace of A and let $\mathcal{Q}(A)$ be the set of m × n real matrices with the same sign pattern as A. If it is true for all $\tilde{A} \in \mathcal{Q}(A)$ that (i) $\Null(\tilde{A}) = \{ \origin \}$, then A is an \define{$L$-matrix}, and if it is true for all $\tilde{A} \in \mathcal{Q}(A)$ that (ii) $\Null(\tilde{A})$ is a line penetrating the open positive orthant of {\Rn}, then A is an S-matrix. When (i) or (ii) holds for each $\tilde{A} \in \mathcal{Q}(A)$ and (ii) applies to at least one, then A is a CS-matrix. It is known that the study of sign-solvability of linear systems may be decomposed into the study of L- and S-matrices, while the study of conditional sign-solvability may be decomposed into that of L- and CS-matrices. When sign-solvability and conditional sign-solvability are regarded as properties of systems of sign cones in {\Rm} associated with the relevant matrices' columns, they have natural generalizations to systems of arbitrary subsets of {\Rm}. This paper contains extended decomposition results for arbitrary systems in the case of sign-solvability, and in the case of conditional sign-solvability, for arbitrary systems of subsets in which each subset is relatively open in its linear span.