## Doctoral Theses

### Exact Order Matrices and the Linear Complimentarity Problem.

February 2011

2-28-2012

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

Mohan, S. R.

#### Abstract (Summary of the Work)

A real n by n matrix M is called an N(P)- matrix of exact order k, if the principal minors of M of order upto (n- k), are negative (positive) and (n - k + 1) to n are positive (negative). In this dissertation, we study the properties of these matrix classes using the linear complementarity problem lep(g, M), for each qe R". Emphasis is placed on N and P-matrices of exact orders 0,1 and 2.Chapter 1 provides the necessary background on linear complementarity and its connection with game theory. Lemke's algorithm is introduced and a brief survey, on some already known classes of matrices in relation to the Icp(g, M) is brought out.A complete characterization of the class of exact order 0 based on the num- ber of solutions to the lep(q, M) for each q â‚¬ R", is presented in chapter 2. Also, a sign reversal property for N-matrices is proved. Counterexample to a well-known characterization on P-matrices is given in the end, while a proof of the same result is provided for the size of the matrix, n <3. Chapter 3 deals with matrices of exact order 1. Here, results on the number of solutions to the lepP(4, M) for each q E R is presented for both the catogories of exact order 1. In the end, a generalizution of exact order one is given, and + a characterization of these matrices in terms of the lep(q, M) is brought out.Chapter 4 is on matrices of exact order 2 or more; we at first define three different categories that evolve in these matrices and study their Q-nature. A complete characterization of the class of exact order 2 and a partial one of the general exact order k are presented. We also look into the following question: When v(M) < 0 and M is of exact order k, can we say that -M E Q? We present a few modifications of the already known algorithms that would process the lep(4, M) when M is a matrix of exact order 2. Also, the difficulties that crop up as we go up the hierarchy in these classes, are cited in the end.C1differentiable maps with the Jacobians being matrices exact order k, are studied in chapter 5. Gale-Nikaido result is extended for C1-maps with Jacobians being exact order k of the first category. Finally, a result on the global univalence of C1-maps when the Jucobian is a matrix of exact order 2 is proved.

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