Berman / Plemmons Nonnegative Matrices in the Mathematical Sciences

1. Auflage 2014
ISBN: 978-1-4832-6086-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 334 Seiten, Web PDF

Nonnegative Matrices in the Mathematical Sciences provides information pertinent to the fundamental aspects of the theory of nonnegative matrices. This book describes selected applications of the theory to numerical analysis, probability, economics, and operations research. Organized into 10 chapters, this book begins with an overview of the properties of nonnegative matrices. This text then examines the inverse-positive matrices. Other chapters consider the basic approaches to the study of nonnegative matrices, namely, geometrical and combinatorial. This book discusses as well some useful ideas from the algebraic theory of semigroups and considers a canonical form for nonnegative idempotent matrices and special types of idempotent matrices. The final chapter deals with the linear complementary problem (LCP). This book is a valuable resource for mathematical economists, mathematical programmers, statisticians, mathematicians, and computer scientists.

Autoren/Hrsg.

Berman, Abraham

Plemmons, Robert J.

Inhaltsverzeichnis

1;Front Cover;1
2;Nonnegative Matrices in the Mathematical Sciences;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;Preface;12
7;Acknowledgments;16
8;Symbols;18
9;Chapter 1. Matrices Which Leave a Cone Invariant;22
9.1;1 Introduction;22
9.2;2 Cones;22
9.3;3 Spectral Properties of Matrices in p(K);27
9.4;4 Cone Primitivity;37
9.5;5 Exercises;40
9.6;6 Notes;44
10;Chapter 2. Nonnegative Matrices;47
10.1;1 Introduction;47
10.2;2 Irreducible Matrices;50
10.3;3 Reducible Matrices;59
10.4;4 Primitive Matrices;66
10.5;5 Stochastic Matrices;69
10.6;6 Exercises;73
10.7;7 Notes;80
11;Chapter 3. Semigroups of Nonnegative Matrices;84
11.1;1 Introduction;84
11.2;2 Algebraic Semigroups;85
11.3;3 Nonnegative Idempotents;85
11.4;4 The Semigroup Nn;88
11.5;5 The Semigroup Nn;103
11.6;6 Exercises;104
11.7;7 Notes;106
12;Chapter 4. Symmetric Nonnegative Matrices;108
12.1;1 Introduction;108
12.2;2 Inverse Eigenvalue Problems;108
12.3;3 Nonnegative Matrices with Given Sums;119
12.4;4 Exercises;127
12.5;5 Notes;130
13;Chapter 5. Generalized Inverse-Positivity;133
13.1;1 Introduction;133
13.2;2 Cone Monotonicity;133
13.3;3 Irreducible Monotonicity;136
13.4;4 Generalized Inverse-Positivity;138
13.5;5 Generalized Monomial Matrices;143
13.6;6 Set Monotonicity;148
13.7;7 Exercises;149
13.8;8 Notes;151
14;Chapter 6. M-Matrices;153
14.1;1 Introduction;153
14.2;2 Nonsingular M-Matrices;154
14.3;3 M-Matrices and Completely Monotonic Functions;163
14.4;4 General M-Matrices;168
14.5;5 Exercises;179
14.6;6 Notes;182
15;Chapter 7. Iterative Methods for Linear Systems;186
15.1;1 Introduction;186
15.2;2 A Simple Example;188
15.3;3 Basic Iterative Methods;191
15.4;4 The SOR Method;193
15.5;5 Nonnegativity and Convergence;201
15.6;6 Singular Linear Systems;216
15.7;7 Exercises;223
15.8;8 Notes;228
16;Chapter 8. Finite Markov Chains;231
16.1;1 Introduction;231
16.2;2 Examples;234
16.3;3 Classical Theory of Chains;238
16.4;4 Modern Analysis of Chains;246
16.5;5 Exercises;257
16.6;6 Notes;261
17;Chapter 9. Input–Output Analysis in Economics;263
17.1;1 Introduction;263
17.2;2 A Simple Application;266
17.3;3 The Open Model;271
17.4;4 The Closed Model;278
17.5;5 Exercises;285
17.6;6 Notes;288
18;Chapter 10. The Linear Complementarity Problem;291
18.1;1 Introduction;291
18.2;2 P-Matrices;292
18.3;3 Q-Matrices;296
18.4;4 Z-Matrices, Least Elements, and Linear Programs;299
18.5;5 Characterizations of Nonsingular M-Matrices;310
18.6;6 Exercises;312
18.7;7 Notes;315
19;References;319
20;Index;334