Positive Definite Matrices

This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry - all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. "Positive Definite Matrices" is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

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