Nonnegative Matrices in the Mathematical Sciences (eBook)

Front Cover 1
Nonnegative Matrices in the Mathematical Sciences 4
Copyright Page 5
Table of Contents 8
Dedication 6
Preface 12
Acknowledgments 16
Symbols 18
Chapter 1. Matrices Which Leave a Cone Invariant 22
1 Introduction 22
2 Cones 22
3 Spectral Properties of Matrices in p(K) 27
4 Cone Primitivity 37
5 Exercises 40
6 Notes 44
Chapter 2. Nonnegative Matrices 47
1 Introduction 47
2 Irreducible Matrices 50
3 Reducible Matrices 59
4 Primitive Matrices 66
5 Stochastic Matrices 69
6 Exercises 73
7 Notes 80
Chapter 3. Semigroups of Nonnegative Matrices 84
1 Introduction 84
2 Algebraic Semigroups 85
3 Nonnegative Idempotents 85
4 The Semigroup Nn 88
5 The Semigroup Nn 103
6 Exercises 104
7 Notes 106
Chapter 4. Symmetric Nonnegative Matrices 108
1 Introduction 108
2 Inverse Eigenvalue Problems 108
3 Nonnegative Matrices with Given Sums 119
4 Exercises 127
5 Notes 130
Chapter 5. Generalized Inverse-Positivity 133
1 Introduction 133
2 Cone Monotonicity 133
3 Irreducible Monotonicity 136
4 Generalized Inverse-Positivity 138
5 Generalized Monomial Matrices 143
6 Set Monotonicity 148
7 Exercises 149
8 Notes 151
Chapter 6. M-Matrices 153
1 Introduction 153
2 Nonsingular M-Matrices 154
3 M-Matrices and Completely Monotonic Functions 163
4 General M-Matrices 168
5 Exercises 179
6 Notes 182
Chapter 7. Iterative Methods for Linear Systems 186
1 Introduction 186
2 A Simple Example 188
3 Basic Iterative Methods 191
4 The SOR Method 193
5 Nonnegativity and Convergence 201
6 Singular Linear Systems 216
7 Exercises 223
8 Notes 228
Chapter 8. Finite Markov Chains 231
1 Introduction 231
2 Examples 234
3 Classical Theory of Chains 238
4 Modern Analysis of Chains 246
5 Exercises 257
6 Notes 261
Chapter 9. Input–Output Analysis in Economics 263
1 Introduction 263
2 A Simple Application 266
3 The Open Model 271
4 The Closed Model 278
5 Exercises 285
6 Notes 288
Chapter 10. The Linear Complementarity Problem 291
1 Introduction 291
2 P-Matrices 292
3 Q-Matrices 296
4 Z-Matrices, Least Elements, and Linear Programs 299
5 Characterizations of Nonsingular M-Matrices 310
6 Exercises 312
7 Notes 315
References 319
Index 334