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Positive Definite Matrices


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Table of Contents

Preface vii Chapter 1: Positive Matrices 1 1.1 Characterizations 1 1.2 Some Basic Theorems 5 1.3 Block Matrices 12 1.4 Norm of the Schur Product 16 1.5 Monotonicity and Convexity 18 1.6 Supplementary Results and Exercises 23 1.7 Notes and References 29 Chapter 2: Positive Linear Maps 35 2.1 Representations 35 2.2 Positive Maps 36 2.3 Some Basic Properties of Positive Maps 38 2.4 Some Applications 43 2.5 Three Questions 46 2.6 Positive Maps on Operator Systems 49 2.7 Supplementary Results and Exercises 52 2.8 Notes and References 62 Chapter 3: Completely Positive Maps 65 3.1 Some Basic Theorems 66 3.2 Exercises 72 3.3 Schwarz Inequalities 73 3.4 Positive Completions and Schur Products 76 3.5 The Numerical Radius 81 3.6 Supplementary Results and Exercises 85 3.7 Notes and References 94 Chapter 4: Matrix Means 101 4.1 The Harmonic Mean and the Geometric Mean 103 4.2 Some Monotonicity and Convexity Theorems 111 4.3 Some Inequalities for Quantum Entropy 114 4.4 Furuta's Inequality 125 4.5 Supplementary Results and Exercises 129 4.6 Notes and References 136 Chapter 5: Positive Definite Functions 141 5.1 Basic Properties 141 5.2 Examples 144 5.3 Loewner Matrices 153 5.4 Norm Inequalities for Means 160 5.5 Theorems of Herglotz and Bochner 165 5.6 Supplementary Results and Exercises 175 5.7 Notes and References 191 Chapter 6: Geometry of Positive Matrices 201 6.1 The Riemannian Metric 201 6.2 The Metric Space Pn 210 6.3 Center of Mass and Geometric Mean 215 6.4 Related Inequalities 222 6.5 Supplementary Results and Exercises 225 6.6 Notes and References 232 Bibliography 237 Index 247 Notation 253

About the Author

Rajendra Bhatia is Professor of Mathematics at the Indian Statistical Institute in New Delhi. He is the author of five books, including Matrix Analysis.


"Written by an expert in the area, the book presents in an accessible manner a lot of important results from the realm of positive matrices and of their applications... The book can be used for graduate courses in linear algebra, or as supplementary material for courses in operator theory, and as a reference book by engineers and researchers working in the applied field of quantum information."--S. Cobzas, Studia Universitatis Babes-Bolyai, Mathematica "There is no obvious competitor for Bhatia's book, due in part to its focus, but also because it contains some very recent material drawn from research articles. Beautifully written and intelligently organised, Positive Definite Matrices is a welcome addition to the literature. Readers who admired his Matrix Analysis will no doubt appreciate this latest book of Rajendra Bhatia."--Douglas Farenick, Image "This is an outstanding book. Its exposition is both concise and leisurely at the same time."--Jaspal Singh Aujla, Zentralblatt MATH

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