A Note on P₁ - and Lipschitzian Matrices

Abstract

The linear complementarity problem (q, A) with data A ∈ R n×n and q ∈ R n involves finding a nonnegative z ∈ R n such that Az + q ≥ 0 and z t (Az + q) = 0. Cottle and Stone introduced the class of P 1-matrices and showed that if A is in P 1 \Q, then K(A) (the set of all q for which (q, A) has a solution) is a half-space and (q, A) has a unique solution for every q in the interior of K(A). Extending the results of Murthy, Parthasarathy, and Sriparna [Ann. Dynamic Games, to appear], we present a number of equivalent characterizations of P 1 \Q. Also, we present yet another characterization of P-matrices. This widens the range of matrix classes for which a conjecture raised by Murthy, Parthasarathy, and Sriparna [SIAM J. Matrix Anal. Appl., 19 (1998), pp. 898-905] characterizing the class of Lipschitzian matrices is true.