Sign Reversing and Matrix Classes

The concept of sign reversing is a useful tool to characterize certain matrix classes in linear complementarity problems. In this paper, we characterize the sign-reversal set of an arbitrary square matrix Μ in terms of the null spaces of the matrices Ι-Λ+ΛΜ, where Λ is a diagonal matrix such that 0≤Λ≤Ι. These matrices are used to characterize the membership of Μ in the classes P₀, P, and the class of column-sufficient matrices. A simple proof of the Gale and Nikaido characterization theorem for the membership in P is presented. We also study the class of diagonally semistable matrices. We prove that this class is contained properly in the class of sufficient matrices. We show that to characterize the diagonally semistable property is equivalent to solving a concave Lagrangian dual problem. For 2 × 2 matrices, there is no duality gap between a primal problem and its Lagrangian problem. Such a primal problem is motivated by the definition of column sufficiency.