A note on P-1- and Lipschitzian matrices

The linear complementarity problem (q, A) with data A is an element of R-nxn and q is an element of R-n involves finding a nonnegative z is an element of R-n such that Az + q greater than or equal to 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.