Nonnegative Matrices, Positive Operators, and Applications

by Ding, Jiu; Zhou, Aihui

• ISBN13: 9789812839176
• ISBN10: 9812839178
• Format: Hardcover
• Copyright: 2009-08-24
• Publisher: World Scientific Pub Co Inc

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
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