Nonnegative Matrices and Applicable Topics in Linear Algebra

✍ Scribed by Alexander Graham

• Publisher: Dover Publications
• Year: 2019
• Tongue: English
• Leaves: 314
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Title Page
Copyright Page
Contents
Preface
Examples of Notation Used
Chapter 1. Introductory Survey
1.1 Introduction
1.2 Notation
1.3 Submatrices and Minors
1.4 Expressing a Singular Matrix as a Product
1.5 Determinants
1.6 The Derivative of a Determinant
1.7 The Characteristic Equation
1.8 The Adjoint of the Characteristic Matrix
1.9 Spectral Decomposition
1.10 Inner Product and Norms
1.11 The Trace Function
1.12 Permutation Matrices and Irreducible Matrices
1.13 Some Aspects of the Theory of Graphs
1.14 Matrix Convergence
Chapter 2. Some Matrix Types
2.1 Introduction
2.2 Unitary Matrices
2.3 Hermitian Matrices
2.4 Normal Matrices
Problems
Chapter 3. Positive Definite Matrices
3.1 Introduction
3.2 Quadratic Forms
3.3 Reductions to a Canonical Form
3.4 A Geometrical Application
Problems
3.5 Positive Definite Matrices
Chapter 4. Nonnegative Matrices
4.1 Introduction
4.2 Terminology and Notation
4.3 The Perron–Frobenius Theorem for a Positive Matrix
4.4 Irreducible Matrices
4.5 Cyclic Matrices
4.6 Reducible Matrices
Problems
Chapter 5. M-Matrices
5.1 Introduction
5.2 Non-singular M-Matrices
5.3 Regular Splitting and Solving Simultaneous Equations
Problems
Chapter 6. Finite Markov Chains and Stochastic Matrices
Chapter 7. Some Applications of Nonnegative Matrices
7.1 Introduction
7.2 Some Genetic Models
7.3 Some Economic Models
7.4 Some Markov Chain Models
Appendix
Solutions to Problems
References
Index