Linear systems with signed solutions

Sign-Solvable Cone-Systems

“…By adding appropriate small positive and negative multiplies of y to each of u, v, and w we see that (2) has solutions with sign patterns …”

“…c) the first entry of Ax is positive. Hence (2) has no solutions with sign patterns (6), (7) or (8). Therefore, without knowing the exact values of a, b, c, d, e, f we can say that the sign patterns of the solutions of (2) are precisely those in (4) and (5).…”

“…Hence C 0 x ¼ B is not sign-solvable. Finally, consider the system Cx ¼ B corresponding to (2), that is, C ¼ ðC 1 , C 2 , C 3 Þ where C 1 , C 2 , C 3 are the cones in (9) and B is the cone fð f , 0Þ T : f > 0g. Then as argued previously, the sign patterns of the solutions to each systemC Cx ¼b b(C C 2 QðCÞ,b b 2 QðBÞ) are independent of the choice ofC C andb b.…”

“…This final example indicates that the definition in [1] for sign-solvable cone-systems is perhaps too restrictive, and should be relaxed to allow for the possibility of more than one given sign pattern. The goal of this paper is to show that the notion of sign-solvability can be relaxed to account for systems like (2), and that the characterizations for sign-solvable cone-systems in [1] can be appropriately extended. To this end, we define the cone-system Cx ¼ B to have signed solutions provided there exists a nonempty set S of sign patterns such that the set of sign patterns of solutions toC Cx ¼b b is S for eachC C 2 QðCÞ andb b 2 B.…”

Matching structure of symmetric bipartite graphs and a generalization of Pólya's problem

“…Totally sign-nonsingular matrices play an important role in the sign-solvability of linear systems of equations [2,13,14,26], linear programming [7], and linear complementarity problems [10]. Total sign-nonsingularity can be recognized in polynomial time by testing sign-nonsingularity of a related symmetric matrix [7].…”