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PART I FINITE NON-NEGATIVE MATRICES 1 CHAPTER 1 Fundamental Concepts and Results in the Theory of Non-negative Matrices 3 1.1 The Perron-Frobenius Theorem for Primitive Matrices 3 1.2 Structure of a General Non-negative Matrix 11 1.3 Irreducible Matrices 18 1.4 Perron-Frobenius Theory for Irreducible Matrices 22 Bibliography and Discussion 25 Exercises 26 CHAPTER 2 Some Secondary Theory with Emphasis on Irreducible Matrices, and Applications 30 2.1 The Equations: (sl - T)x = c 30 Bibliography and Discussion to 2.1 38 Exercises on 2.1 39 2.2 Iterative Methods for Solution of Certain Linear Equation Systems 41 Bibliography and Discussion to 2.2 44 Exercises on 2.2 44 2.3 Some Extensions of the Perron-Frobenius Structure 45 Bibliography and Discussion to 2.3 53 Exercises on 2.3 54 2.4 Combinatorial Properties 55 Bibliography and Discussion to 2.4 60 Exercises on 2.4 60 2.5 Spectrum Localization 61 Bibliography and Discussion to 2.5 64 Exercises on 2.5 66 2.6 Estimating Non-negative Matrices from Marginal Totals 67 Bibliography and Discussion to 2.6 75 Exercises on 2.6 79 CHAPTER 3 Inhomogeneous Products of Non-negative Matrices 80 3.1 Birkhoff's Contraction Coefficient: Generalities 80 3.2 Results on Weak Ergodicity 85 Bibliography and Discussion to 3.1-3.2 88 Exercises on 3.1-3.2 90 3.3 Strong Ergodicity for Forward Products 92 Bibliography and Discussion to 3.3 99 Exercises on 3.3 100 3.4 Birkhoff's Contraction Coefficient: Derivation of Explicit Form 100 Bibliography and Discussion to 3.4 111 Exercises on 3.4 111 CHAPTER 4 Markov Chains and Finite Stochastic Matrices 112 4.1 Markov Chains 113 4.2 Finite Homogeneous Markov Chains 118 Bibliography and Discussion to 4.1-4.2 131 Exercises on 4.2 132 4.3 Finite Inhomogeneous Markov Chains and Coefficients of Ergodicity 134 4.4 Sufficient Conditions for Weak Ergodicity 140 Bibliography and Discussion to 4.3-4.4 144 Exercises on 4.3-4.4 147 4.5 Strong Ergodicity for Forward Products 149 Bibliography and Discussion to 4.5 151 Exercises on 4.5 152 4.6 Backwards Products 153 Bibliography and Discussion to 4.6 157 Exercises on 4.6 158 PART II COUNTABLE NON-NEGATIVE MATRICES 159 CHAPTER 5 Countable Stochastic Matrices 161 5.1 Classification of Indices 161 5.2 Limiting Behaviour for Recurrent Indices 168 5.3 Irreducible Stochastic Matrices 172 5.4 The "Dual" Approach; Subinvariant Vectors 5.5 Potential and Boundary Theory for Transient Indices 181 5.6 Example 191 Bibliography and Discussion 194 Exercises 195 CHAPTER 6 Countable Non-negative Matrices 199 6.1 The Convergence Parameter R, and the R-Classification of T 200 6.2 R-Subinvariance and Invariance; R-Positivity 205 6.3 Consequences for Finite and Stochastic Infinite Matrices 207 6.4 Finite Approximations to Infinite Irreducible T 210 6.5 An Example 215 Bibliography and Discussion 218 Exercises 219 CHAPTER 7 Truncations of Infinite Stochastic Matrices 221 7.1 Determinantal and Cofactor Properties 222 7.2 The Probability Algorithm 229 7.3 Quasi-stationary Distributions 236 Bibliography and Discussion 242 Exercises 242 APPENDICES 245 Appendix A. Some Elementary Number Theory 247 Appendix B. Some General Matrix Lemmas 252 Appendix C. Upper Semi-continuous Functions 255 Bibliography 257

Library of Congress subject headings for this publication: Non-negative matrices, Markov processes, Matrices non négatives, Markov, Processus de