Nonnegative Matrices and Applicable Topics in Linear Algebra

by Alexander Graham

Overview

Nonnegative matrices is an increasingly important subject in economics, control theory, numerical analysis, Markov chains, and other areas. This concise treatment is directed toward undergraduates who lack specialized knowledge at the postgraduate level of mathematics and related fields, such as mathematical economics and operations research.
An Introductory Survey encompasses some aspects of matrix theory and its applications and other relevant topics in linear algebra, including certain facets of graph theory. Subsequent chapters cover various points of the theory of normal matrices, comprising unitary and Hermitian matrices, and the properties of positive definite matrices. An exploration of the main topic, nonnegative matrices, is followed by a discussion of M-matrices. The final chapter examines stochastic, genetic, and economic models. The important concepts are illustrated by simple worked examples. Problems appear at the conclusion of most chapters, with solutions at the end of the book.

Product Details

ISBN-13:	9780486838076
Publisher:	Dover Publications
Publication date:	11/13/2019
Series:	Dover Books on Mathematics
Edition description:	Reprint
Pages:	272
Sales rank:	676,597
Product dimensions:	6.00(w) x 8.90(h) x 0.60(d)

About the Author

Alexander Graham, now retired, was Senior Lecturer on the Faculty of Mathematics at The Open University, Milton Keynes, U.K. Dover also publishes his Matrix Theory and Applications for Scientists and Engineers, and Kronecker Products and Matrix Calculus with Applications.

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PREFACE

Although a number of books making reference to the theory of nonnegative matrices exist, most of them treat this topic at a rather sophisticated postgraduate level. The recent rapid expansion in the development and application of various aspects of economic, stochastic and mathematical disciplines which make use of this theory has persuaded me that there is a need for a text which can be understood by readers who are not necessarily expert mathematicians.

To achieve this aim, I soon realised that it would also be necessary to write a background and an introductory survey which included some aspects of matrix theory which may not be well known to the potential reader nor particularly easily found in the required form in the current literature. Chapter 1 is a collection of various topics in linear algebra and closely related matter most of which are used in various forms in the development of the theory of nonnegative matrices. The kernel of this theory is the Perron — Frobenius theorem dealing with the special properties of a matrix having nonnegative entries. Since there is little point in investigating the characteristics of one family of matrices, however simple and beautiful they may be, without an investigation of the properties of some of the other families of matrices, Chapters 2 and 3 are written to do just that. The main theme of the book, aspects of the theory of nonnegative matrices, is discussed in Chapter 4. Chapters 5 and 6 can be regarded as applications of the theory set out in Chapter 4.

Most of the important concepts in the text are illustrated by simple worked examples and Problems at the end of each chapter (except Chapters 1 and 6) with solutions at the end of the book.

Now for a brief outline of the content of the book.

Chapter 1 may be regarded as an introduction to various properties of matrices and determinants. It includes a section on permutation matrices and, in contrast, a section on the theory of graphs, both topics very helpful in the development of Chapter 4. Chapter 2 covers various aspects of the theory of Normal matrices, including unitary and Hermitian matrices. Chapter 3 is devoted to the discussion of some of the properties of positive definite matrices.

Chapter 4 covers the main topic in this book, nonnegative matrices. Section 4.3 deals mainly with Positive matrices and some of the arguments used are repeated, even if in a modified form, when discussing other classes of Irreducible matrices in Sections 4.4 and 4.5. The policy of repeating some of the arguments is intentional and was dictated by my desire to make this topic as comprehensible as possible to the reader. Section 4.6 covers various properties of reducible matrices.

A discussion of M-matrices will be found in Chapter 5. It can be considered as a mathematical application of the results proved in Chapter 4. Finally Chapter 6 is a direct application of these results to Markov chains and stochastic matrices. Obviously stochastic matrices form a subclass of the family of nonnegative matrices and much of the development of this topic is due to our interest in Stochastic matrices.

Worked examples occur throughout the book and serve various purposes; some illustrate important concepts whereas others are applications of these concepts to linear algebra. The applications presented in Chapters 6 and 7 are devoted to various applications of the theory which was developed in Chapters 1 to 5. The final chapter considers a variety of applications and investigations emphasising the great importance of nonnegative matrices and includes some genetic, economic and Markov chain models.

References are given in two forms; [BX] is to book number X in the book reference section and [Y] is the paper number Y in the paper reference section.

(Continues…)

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Excerpted from "Nonnegative Matrices & Applicable Topics Linear Algebra"
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Copyright © 1987 A. Graham.
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Table of Contents

CONTENTS

Preface
Examples of Notation Used
Chapter 1. Introductory Survey
Chapter 2. Some Matrix Types
Chapter 3. Positive Definite Matrices
Chapter 4. Nonnegative Matrices
Chapter 5. M-Matrices
Chapter 6. Finite Markov Chains and Stochastic Matrices
Chapter 7. Some Applications of Nonnegative Matrices
Appendix
Solutions to Problems
References
Index