A Note on Murthy's Conjecture

Abstract

Abstract. In this paper, we consider a conjecture made by Murthy to the effect that a CQ f"l Qo matrix is positive semidefinite (PSD) and show that the conjecture is true for n x n matrices of rank 1 or 3 x 3 matrices that are not in Q. We also consider the class of Pi-matrices which is a subclass of Qo and obtain the following: for A 6 Pi , if A S CQ and A £ Q, then A is PSD; if A 6 C{ and A € Q, then A is PSD for n < 3 and A isn't PSD for n > 3. IntroductionThe linear complementarity problem (LCP) with data A G R nxn and q € R n involves finding a vector z E R n such thatLCP has numerous applications, both in theory and practice, treated by vast literature (see [l]). A number of matrix classes have been defined in connection with LCP. We shall briefly introduce the concepts and notation required for presentation of the results of this paper.For most of other notion, we shall follow that used in [1]. For any positive integer n, write n = {1,2, • • •, n}, and for any subset a of n, write a = n\a. Consider A £ R nxn .If a C n such that detA aa ^ 0, then the matrix M defined byis known as the principal pivotal transform (PPT) of A with respect to a This work is supported by YSF of Guangdong University of Technology (062058)