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Positive Definite Matrices
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Positive Definite Matrices
by Rajendra Bhatia
Positive Definite Matrices
Cover Page
Title Page
Copyright Page
Contents
Preface
Chapter One: Positive Matrices
1.1 Characterizations
1.2 Some Basic Theorems
1.3 Block Matrices
1.4 Norm of the Schur Product
1.5 Monotonicity and Convexity
1.6 Supplementary Results and Exercises
1.7 Notes and References
Chapter Two: Positive Linear Maps
2.1 Representations
2.2 Positive Maps
2.3 Some Basic Properties of Positive Maps
2.4 Some Applications
2.5 Three Questions
2.6 Positive Maps on Operator Systems
2.7 Supplementary Results and Exercises
2.8 Notes and References
Chapter Three: Completely Positive Maps
3.1 Some Basic Theorems
3.2 Exercises
3.3 Schwarz Inequalities
3.4 Positive Completions and Schur Products
3.5 The Numerical Radius
3.6 Supplementary Results and Exercises
3.7 Notes and References
Chapter Four: Matrix Means
4.1 The Harmonic Mean and the Geometric Mean
4.2 Some Monotonicity and Convexity Theorems
4.3 Some Inequalities for Quantum Entropy
4.4 Furuta's Inequality
4.5 Supplementary Results and Exercises
4.6 Notes and References
Chapter Five: Positive Definite Functions
5.1 Basic Properties
5.2 Examples
5.3 Loewner Matrices
5.4 Norm Inequalities for Means
5.5 Theorems of Herglotz and Bochner
5.6 Supplementary Results and Exercises
5.7 Notes and References
Chapter Six: Geometry of Positive Matrices
6.1 The Riemannian Metric
6.2 The Metric Space
6.3 Center of Mass and Geometric Mean
6.4 Related Inequalities
6.5 Supplementary Results and Exercises
6.6 Notes and References
Bibliography
Index
Notation
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