Doctoral Theses

Some Contributions to Generalized Inverse and the Linear Complementarity Problem.

5-22-1989

5-22-1990

Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

Degree Name

Doctor of Philosophy

Mathematics

Department

SQC and OR Unit (Bangalore)

Supervisor

Ramamurthy, K. G. (SQCOR-Bangalore; ISI)

Abstract (Summary of the Work)

A generalized inverse (g-inverse) of a matrix A is a solution x to the matrix equationA XA = A(1.1.1)A g-inverse of A can be defined alternatively as a matrix x such that x = Xb is a solution to the linear equation Ax -b for any b that makes - b consistent. There is a vast literature on g-inverse. For a number of results on g-inverses and their applications one may refer to the well known books in the literature by Rao and Mitra (1971); and by Ben Israel and Greville (1974).Another inverse that lies hidden in the definition of g-inverse is outer inverse. An outer inverse of a matrix A is a solution x to the matrix equationX A X - X(1.1.2)Ben Israel and Greville (1974) give some applications of outer inverse. A recent book by Getson and Hsuan (1988) lays emphasis on outer inverses and highlights their role in statistical applications.Unless A is nonsingular, a g-inwerse of A is not unique. Similarly an outer inverse of A is not unique unlesS A = 0. This has led to the introduction of a variety of inverses in the literature for various applications. How ever we have several results on characterization of these inverses available to us. These results enable us to understand the key vari ables that give rise to different types of invers es. Usually in the X literature g-iwerses and outer inverses are treated separately. In this thesis we introduce an integrated approach for studying both g-inverses and outer-inverses. This we accomplish by means of bordered matrices. Also we derive some n ew results using the new approach.5Given an nxn real matrix M and an n-dimensional real vector q, the linear complementarity problem (LCP) is a problem of computing a solution (w ,z), if it exists, to the following system of equations:

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