In this manuscript, we present some new results for the semidefinite linear complementarity problem in the context of three notions for linear transformations, viz., pseudo w-P property, pseudo Jordan w-P property and pseudo SSM property. | Yugoslav Journal of Operations Research 27 (2017), Number 2, 135–151 DOI: EXTENSIONS OF P-PROPERTY, R0 -PROPERTY AND SEMIDEFINITE LINEAR COMPLEMENTARITY PROBLEMS I. JEYARAMAN Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal - 575 025, India i jeyaraman@ Kavita BISHT Department of Mathematics,Indian Institute of Technology Madras, Chennai - 600 036, India kavitabishtiitm2512@ . SIVAKUMAR Department of Mathematics,Indian Institute of Technology Madras, Chennai - 600 036, India kcskumar@ Received: January 2017 / Accepted: May 2017 Abstract: In this manuscript, we present some new results for the semidefinite linear complementarity problem in the context of three notions for linear transformations, viz., pseudo w-P property, pseudo Jordan w-P property, and pseudo SSM property. Interconnections with the P# -property (proposed recently in the literature) are presented. We also study the R# -property of a linear transformation, extending the rather well known notion of an R0 -matrix. In particular, results are presented for the Lyapunov, Stein, and the multiplicative transformations. Keywords: Linear Complementarity Problem, P-property, R-property, Semidefinite Linear Complementarity Problem, w-P properties, Jordan w-P property, Moore-Penrose Inverse. MSC: 90C33, 15A09. 136 , , / Extensions of P-property 1. INTRODUCTION Let Sn denote the vector space of all n × n real symmetric matrices and Sn+ be the set of all symmetric positive semidefinite matrices in Sn . Given a linear transformation L : Sn → Sn and a matrix Q ∈ Sn , the semidefinite linear complementarity problem, denoted by SDLCP(L, Q), is to find an X ∈ Sn such that X ∈ Sn+ , Y = L(X) + Q ∈ Sn+ , and hX, Yi = tr(XY) = 0, where tr(A) denotes the trace of the square matrix A. If such an X exists, we call X, a solution of SDLCP(L, Q). The set of all
