Nonnegative Matrices, Positive Operators, and Applications
by Ding, Jiu; Zhou, AihuiFormat: Hardcover
Pub. Date: 2009-08-24
Publisher(s): World Scientific Pub Co Inc
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Summary
Nonnegative matrices and positive operators are widely applied in science, engineering, and technology. This book provides the basic theory and several typical modern science and engineering applications of nonnegative matrices and positive operators, including the fundamental theory, methods, numerical analysis, and applications in the Google search engine, computational molecular dynamics, and wireless communications. Unique features of this book include the combination of the theories of nonnegative matrices and positive operators as well as the emphasis on applications of nonnegative matrices in the numerical analysis of positive operators, such as Markov operators and FrobeniusPerron operators both of which play key roles in the statistical and stochastic studies of dynamical systems. It can be used as a textbook for an upper level undergraduate or beginning graduate course in advanced matrix theory and/or positive operators as well as for an advanced topics course in operator theory or ergodic theory. In addition, it serves as a good reference for researchers in mathematical sciences, physical sciences, and engineering.
Table of Contents
| Preface | p. vii |
| Elementary Properties of Nonnegative Matrices | p. 1 |
| Algebraic Properties of Nonnegative Matrices | p. 1 |
| Spectral Properties of Nonnegative Matrices | p. 4 |
| Combinatorial Properties of Nonnegative Matrices | p. 5 |
| M-matrices | p. 10 |
| Notes and Remarks | p. 15 |
| Exercises | p. 16 |
| The Perron-Frobenius Theory for Nonnegative Matrices | p. 19 |
| Perron's Theorem for Positive Matrices | p. 19 |
| Frobenius' Theorem for Irreducible Nonnegative Matrices | p. 26 |
| Ergodic Theorems for Nonnegative Matrices | p. 35 |
| Characterizations of Irreducibility and Primitivity | p. 40 |
| Notes and Remarks | p. 46 |
| Exercises | p. 47 |
| Stochastic Matrices | p. 51 |
| Stochastic Matrices | p. 51 |
| Doubly Stochastic Matrices | p. 57 |
| Inequalities Related to Doubly Stochastic Matrices | p. 65 |
| Ergodic Theorems of Stochastic Matrices | p. 69 |
| Notes and Remarks | p. 71 |
| Exercises | p. 72 |
| Applications of Nonnegative Matrices | p. 75 |
| Finite State Stationary Markov Chains | p. 75 |
| The Web Information Retrieval | p. 81 |
| Discrete Dynamical Geometry | p. 85 |
| Iterative Solutions of Linear Algebraic Equations | p. 90 |
| Notes and Remarks | p. 93 |
| Exercises | p. 94 |
| Fundamental Theory of Positive Operators | p. 97 |
| Banach Lattices | p. 98 |
| Positive Operators | p. 106 |
| Ideals and Projections | p. 114 |
| L-spaces and M-spaces | p. 118 |
| Notes and Remarks | p. 124 |
| Exercises | p. 125 |
| The Spectral Theory of Positive Operators | p. 127 |
| The Spectral Theory of Bounded Linear Operators | p. 127 |
| The Spectral Theory of Positive Operators | p. 137 |
| Ergodic Theorems of Bounded Linear Operators | p. 142 |
| Ergodic Theorems of Positive Operators | p. 146 |
| Notes and Remarks | p. 150 |
| Exercises | p. 152 |
| Markov Operators | p. 155 |
| Definition and Examples of Markov Operators | p. 155 |
| Properties of Markov Operators | p. 161 |
| Ergodic Theorems of Markov Operators | p. 164 |
| Asymptotic Periodicity of Markov Operators | p. 168 |
| Notes and Remarks | p. 177 |
| Exercises | p. 178 |
| Frobenius-Perron Operators | p. 181 |
| Measure-preserving Transformations | p. 181 |
| Frobenius-Perron Operators | p. 184 |
| Existence of Absolutely Continuous Invariant Measures | p. 191 |
| Ergodic Properties of Invariant Measures | p. 199 |
| Notes and Remarks | p. 207 |
| Exercises | p. 208 |
| Composition Operators | p. 211 |
| Positive Functionals on C(K) | p. 211 |
| Invariant Measures of Continuous Transformations | p. 213 |
| Ergodic Theorems of Composition Operators | p. 216 |
| Topological Transitivity and Ergodicity | p. 223 |
| Notes and Remarks | p. 226 |
| Exercises | p. 227 |
| Approximations of Markov Operators | p. 229 |
| Ulam's Piecewise Constant Approximations | p. 230 |
| Piecewise Linear Markov Approximations | p. 237 |
| Convergence Rate of Ulam's Method | p. 246 |
| Convergence Rate of the Markov Method | p. 252 |
| Notes and Remarks | p. 259 |
| Exercises | p. 259 |
| Applications of Positive Operators | p. 261 |
| Positive Approximations of Continuous Functions | p. 261 |
| Evolution Differential Equations | p. 268 |
| Conformational Molecular Dynamics | p. 272 |
| Third Generation Wireless Communications | p. 283 |
| Notes and Remarks | p. 288 |
| Exercises | p. 288 |
| Foundations of Matrix Theory | p. 291 |
| Basic Concepts and Results | p. 291 |
| The Spectral Theory of Matrices | p. 294 |
| Analytic Theory of Matrices | p. 301 |
| Norms and Convergence | p. 304 |
| Basic Operator Theory | p. 307 |
| Banach Spaces | p. 307 |
| Hilbert Spaces | p. 313 |
| Bounded Linear Operators | p. 315 |
| Bounded Linear Functionals and Dual Operators | p. 319 |
| Bibliography | p. 327 |
| Index | p. 333 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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