On “P” property and the column-W property

Abstract

P-matrices play an important role in the well-posedness of a linear complementarity problem (LCP). Similarly, the well-posedness of a horizontal linear complementarity problem (HLCP) is closely related to the column-W property of a matrix k-tuple.In this paper we first consider the problem of generating P-matrices from a given pair of matrices. Given a matrix pair (D,F) where D is a square matrix of order m and matrix F has m rows, “what are the conditions under which there exists a matrix G such that (D+FG) is a P-matrix?”. We obtain necessary and sufficient conditions for the special case when the column rank of F is m−1. A decision algorithm of complexity O(m2) to check whether the given pair of matrices (D,F) is P-matrisable is obtained. We also obtain a necessary and an independent sufficient condition for the general case when rank(F) is less than m−1.We then generalise the P-matrix generating problem to the generation of matrix k-tuples satisfying the column-W property from a given matrix (k+1)-tuple. That is, given a matrix (k+1)-tuple (D1,…,Dk,F), where Djs are square matrices of order m and F is a matrix having m rows, we determine the conditions under which the matrix k-tuple (D1+FG1,…,Dk+FGk) satisfies the column-W property. As in the case of P-matrices we obtain necessary and sufficient conditions for the case when rank(F)=m−1. Using these conditions a decision algorithm of complexity O(km2) to check whether the given matrix (k+1)-tuple is column-W matrisable is obtained. Then for the case when rank(F) is less than m−1, we obtain a necessary and an independent sufficient condition.For a special sub-class of P-matrices we give a polynomial time decision algorithm for P-matrisability. Finally, we obtain a geometric characterisation of column-W property by generalising the well known separation theorem for P-matrices.