Positive Definite Matrices

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