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Positive Definite Matrices
Bhatia, Rajendra, 1952-
Copyright2007 by Princeton University Press
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
vi CONTENTS
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
Preface
The theory of positive definite matrices, positive definite functions,
and positive linear maps is rich in content. It offers many beautiful
theorems that are simple and yet striking in their formulation, uncomplicated
and yet ingenious in their proof, diverse as well as powerful
in their application. The aim of this book is to present some of these
results with a minimum of prerequisite preparation.
The seed of this book lies in a cycle of lectures I was invited to give
at the Centro de Estruturas Lineares e Combinat´orias (CELC) of the
University of Lisbon in the summer of 2001. My audience was made
up of seasoned mathematicians with a distinguished record of research
in linear and multilinear algebra, combinatorics, group theory, and
number theory. The aim of the lectures was to draw their attention
to some results and methods used by analysts. A preliminary draft
of the first four chapters was circulated as lecture notes at that time.
Chapter 5 was added when I gave another set of lectures at the CELC
in 2003.
Because of this genesis, the book is oriented towards those interested
in linear algebra and matrix analysis. In some ways it supplements
my earlier book Matrix Analysis (Springer, Graduate Texts in Mathematics,
Volume 169). However, it can be read independently of that
book. The usual graduate-level preparation in analysis, functional
analysis, and linear algebra provides adequate background needed for
reading this book.