# Nonnegative Matrices in the Mathematical Sciences

## 1st Edition

Authors:
eBook ISBN: 9781483260860
Imprint: Academic Press
Published Date: 1st January 1979
Page Count: 334
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## Description

Nonnegative Matrices in the Mathematical Sciences provides information pertinent to the fundamental aspects of the theory of nonnegative matrices. This book describes selected applications of the theory to numerical analysis, probability, economics, and operations research.

Organized into 10 chapters, this book begins with an overview of the properties of nonnegative matrices. This text then examines the inverse-positive matrices. Other chapters consider the basic approaches to the study of nonnegative matrices, namely, geometrical and combinatorial. This book discusses as well some useful ideas from the algebraic theory of semigroups and considers a canonical form for nonnegative idempotent matrices and special types of idempotent matrices. The final chapter deals with the linear complementary problem (LCP).

This book is a valuable resource for mathematical economists, mathematical programmers, statisticians, mathematicians, and computer scientists.

## Table of Contents

Preface

Acknowledgments

Symbols

Chapter 1 Matrices Which Leave a Cone Invariant

1 Introduction

2 Cones

3 Spectral Properties of Matrices in π(K)

4 Cone Primitivity

5 Exercises

6 Notes

Chapter 2 Nonnegative Matrices

1 Introduction

2 Irreducible Matrices

3 Reducible Matrices

4 Primitive Matrices

5 Stochastic Matrices

6 Exercises

7 Notes

Chapter 3 Semigroups of Nonnegative Matrices

1 Introduction

2 Algebraic Semigroups

3 Nonnegative Idempotents

4 The Semigroup Nn

5 The Semigroup Dn

6 Exercises

7 Notes

Chapter 4 Symmetric Nonnegative Matrices

1 Introduction

2 Inverse Eigenvalue Problems

3 Nonnegative Matrices with Given Sums

4 Exercises

5 Notes

Chapter 5 Generalized Inverse-Positivity

1 Introduction

2 Cone Monotonicity

3 Irreducible Monotonicity

4 Generalized Inverse-Positivity

5 Generalized Monomial Matrices

6 Set Monotonicity

7 Exercises

8 Notes

Chapter 6 M-Matrices

1 Introduction

2 Nonsingular M-Matrices

3 M-Matrices and Completely Monotonic Functions

4 General M-Matrices

5 Exercises

6 Notes

Chapter 7 Iterative Methods for Linear Systems

1 Introduction

2 A Simple Example

3 Basic Iterative Methods

4 The SOR Method

5 Nonnegativity and Convergence

6 Singular Linear Systems

7 Exercises

8 Notes

Chapter 8 Finite Markov Chains

1 Introduction

2 Examples

3 Classical Theory of Chains

4 Modern Analysis of Chains

5 Exercises

6 Notes

Chapter 9 Input-Output Analysis in Economics

1 Introduction

2 A Simple Application

3 The Open Model

4 The Closed Model

5 Exercises

6 Notes

Chapter 10 The Linear Complementarity Problem

1 Introduction

2 P-Matrices

3 Q-Matrices

4 Z-Matrices, Least Elements, and Linear Programs

5 Characterizations of Nonsingular M-Matrices

6 Exercises

7 Notes

References

Index

## Details

No. of pages:
334
Language:
English
Copyright:
© Academic Press 1979
Published:
Imprint:
Academic Press
eBook ISBN:
9781483260860