A NOTE ON P1- AND LIPSCHITZIAN MATRICES

Abstract. The linear complementarity problem (q;A) with data A 2 Rnn and q 2 Rn involves nding a nonnegative z 2 Rn such that Az+ q 0 and zt(Az+ q) = 0. Cottle and Stone introduced the class of P1-matrices and showed that if A is in P1nQ, 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 P1nQ. 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