G. S. R. Murthy

Lipschitzian Q -matrices are P -matrices

In this note, we show that LipschitzianQ-matrices areP-matrices by obtaining a necessary condition on LipschitzianQ0-matrices. The sufficiency of this condition has also been established by the first two authors along with another coauthor (Murthy, Parthasarathy and Sriparna, 1995).

Fully copositive matrices

The class of fully copositive (C0f) matrices introduced in [G.S.R. Murthy, T. Parthasarathy, SIAM Journal on Matrix Analysis and Applications 16 (4) (1995) 1268–1286] is a subclass of fully semimonotone matrices and contains the class of positive semidefinite matrices. It is shown that fully copositive matrices within the class ofQ0-matrices areP0-matrices. As a corollary of this main result, we establish that a bisymmetricQ0-matrix is positive semidefinite if, and only if, it is fully copositive. Another important result of the paper is a constructive characterization ofQ0-matrices within the class ofC0f. While establishing this characterization, it will be shown that Gravess principal pivoting method of solving Linear Complementarity Problems (LCPs) with positive semidefinite matrices is also applicable toC0f ∩Q0 class. As a byproduct of this characterization, we observe that aC0f-matrix is inQ0 if, and only if, it is completelyQ0. Also, from Aganagic and Cottles [M. Aganagic, R.W. Cottle, Mathematical Programming 37 (1987) 223–231] result, it is observed that LCPs arising fromC0f ∩Q0 class can be processed by Lemkes algorithm.

A copositive Q -matrix which is not R 0

Jeter and Pye gave an example to show that Pangs conjecture, thatL1 ⋂Q ⊂R0, is false while Seetharama Gowda showed that the conjecture is true for symmetric matrices. It is known thatL1-symmetric matrices are copositive matrices. Jeter and Pye as well as Seetharama Gowda raised the following question: Is it trueC0 ⋂Q ⊂R0? In this note we present an example of a copositive Q-matrix which is notR0. The example is based on the following elementary proposition: LetA be a square matrix of ordern. SupposeR1 =R2 whereRi stands for theith row ofA. Further supposeA11 andA22 are Q-matrices whereAii stands for the principal submatrix omitting theith row andith column fromA. ThenA is a Q-matrix.

Some Recent Results on The Linear Complementarity Problem

In this article we present some recent results on the linear complementarity problem. It is shown that (i) within the class of column adequate matrices, a matrix is in

Constructive characterization of Lipschitzian Q0-matrices

\Qnot

On the Solution Sets of Linear Complementarity Problems

if and only if it is completely

On co-positive, semi-monotone Q -matrices

\Qnot

ON FULLY SEMIMONOTONE MATRICES

(ii) for the class of

Optimization in PCB Manufacturing

\Cnotf

Statistics for Measuring Tyre Contact Patch

-matrices introduced by Murthy and Parthasarathy [SIAM J. Matrix Anal. Appl., 16 (1995), pp. 1268--1286], we provide a sufficient condition under which a matrix is in