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On "P" property and the column-W property

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Title On "P" property and the column-W property
 
Creator PRIYADARSHAN, H
PILLAI, HK
 
Subject P-matrix
Column-W property
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Well-posedness
Regularisation of linear complementarity systems
Regularisation of piecewise linear systems
Separation theorem
HYBRID SYSTEMS
MATRICES
 
Description 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(m(2)) 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 (D-1,D- ... , D-k, F), where D(j)s are square matrices of order m and F is a matrix having m rows, we determine the conditions under which the matrix k-tuple (D-1 + FG(1), ... , D-k + FG(k)) 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(km(2)) 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. (C) 2011 Elsevier Inc. All rights reserved.
 
Publisher ELSEVIER SCIENCE INC
 
Date 2014-10-15T12:45:33Z
2014-10-15T12:45:33Z
2012
 
Type Article
 
Identifier LINEAR ALGEBRA AND ITS APPLICATIONS, 436(7)1969-1989
http://dx.doi.org/10.1016/j.laa.2011.10.034
http://dspace.library.iitb.ac.in/jspui/handle/100/14921
 
Language en