Positive Definite Matrices

Book Description

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

  1. Cover Page
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Preface
  6. Chapter One: Positive Matrices
    1. 1.1 Characterizations
    2. 1.2 Some Basic Theorems
    3. 1.3 Block Matrices
    4. 1.4 Norm of the Schur Product
    5. 1.5 Monotonicity and Convexity
    6. 1.6 Supplementary Results and Exercises
    7. 1.7 Notes and References
  7. Chapter Two: Positive Linear Maps
    1. 2.1 Representations
    2. 2.2 Positive Maps
    3. 2.3 Some Basic Properties of Positive Maps
    4. 2.4 Some Applications
    5. 2.5 Three Questions
    6. 2.6 Positive Maps on Operator Systems
    7. 2.7 Supplementary Results and Exercises
    8. 2.8 Notes and References
  8. Chapter Three: Completely Positive Maps
    1. 3.1 Some Basic Theorems
    2. 3.2 Exercises
    3. 3.3 Schwarz Inequalities
    4. 3.4 Positive Completions and Schur Products
    5. 3.5 The Numerical Radius
    6. 3.6 Supplementary Results and Exercises
    7. 3.7 Notes and References
  9. Chapter Four: Matrix Means
    1. 4.1 The Harmonic Mean and the Geometric Mean
    2. 4.2 Some Monotonicity and Convexity Theorems
    3. 4.3 Some Inequalities for Quantum Entropy
    4. 4.4 Furuta's Inequality
    5. 4.5 Supplementary Results and Exercises
    6. 4.6 Notes and References
  10. Chapter Five: Positive Definite Functions
    1. 5.1 Basic Properties
    2. 5.2 Examples
    3. 5.3 Loewner Matrices
    4. 5.4 Norm Inequalities for Means
    5. 5.5 Theorems of Herglotz and Bochner
    6. 5.6 Supplementary Results and Exercises
    7. 5.7 Notes and References
  11. Chapter Six: Geometry of Positive Matrices
    1. 6.1 The Riemannian Metric
    2. 6.2 The Metric Space
    3. 6.3 Center of Mass and Geometric Mean
    4. 6.4 Related Inequalities
    5. 6.5 Supplementary Results and Exercises
    6. 6.6 Notes and References
  12. Bibliography
  13. Index
  14. Notation

Product Information

  • Title: Positive Definite Matrices
  • Author(s): Rajendra Bhatia
  • Release date: January 2009
  • Publisher(s): Princeton University Press
  • ISBN: 9781400827787