Positive Definite Matrices - Rajendra Bhatia 2007

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.

Review:

"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

Endorsement:

"This is a monograph for mathematicians interested in an important realm of matrix-analytic ideas. Like the author's distinguished book, Matrix Analysis, it will be a convenient and much-quoted reference source. There are many wonderful insights in a first-rate exposition of important ideas not easily extracted from other sources. The scholarship is impeccable."--Roger A. Horn, University of Utah

"I believe that every expert in matrix analysis can find something new in this book. Bhatia presents some important material in several topics related to positive definite matrices including positive linear maps, completely positive maps, matrix means, positive definite functions, and geometry of positive definite matrices. There are many beautiful results, useful techniques, and ingenious ideas here. Bhatia's writing style has always been concise, clear, and illuminating."--Xingzhi Zhan, East China Normal University

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

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