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Matrix Completions, Moments, and Sums of Hermitian Squares

Book Description

Intensive research in matrix completions, moments, and sums of Hermitian squares has yielded a multitude of results in recent decades. This book provides a comprehensive account of this quickly developing area of mathematics and applications and gives complete proofs of many recently solved problems. With MATLAB codes and more than 200 exercises, the book is ideal for a special topics course for graduate or advanced undergraduate students in mathematics or engineering, and will also be a valuable resource for researchers.

Often driven by questions from signal processing, control theory, and quantum information, the subject of this book has inspired mathematicians from many subdisciplines, including linear algebra, operator theory, measure theory, and complex function theory. In turn, the applications are being pursued by researchers in areas such as electrical engineering, computer science, and physics.

The book is self-contained, has many examples, and for the most part requires only a basic background in undergraduate mathematics, primarily linear algebra and some complex analysis. The book also includes an extensive discussion of the literature, with close to 600 references from books and journals from a wide variety of disciplines.

Table of Contents

  1. Cover
  2. Half title
  3. Title
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Chapter 1. Cones of Hermitian matrices and trigonometric polynomials
    1. 1.1 Cones and their basic properties
    2. 1.2 Cones of Hermitian matrices
    3. 1.3 Cones of trigonometric polynomials
    4. 1.4 Determinant and entropy maximization
    5. 1.5 Semidefinite programming
    6. 1.6 Exercises
    7. 1.7 Notes
  9. Chapter 2. Completions of positive semidefinite operator matrices
    1. 2.1 Positive definite completions: the banded case
    2. 2.2 Positive definite completions: the chordal case
    3. 2.3 Positive definite completions: the Toeplitz case
    4. 2.4 The Schur complement and Fejér-Riesz factorization
    5. 2.5 Schur parameters
    6. 2.6 The central completion, maximum entropy, and inheritance principle
    7. 2.7 The Hamburger moment problem and spectral factorization on the real line
    8. 2.8 Linear prediction
    9. 2.9 Exercises
    10. 2.10 Notes
  10. Chapter 3. Multivariable moments and sums of Hermitian squares
    1. 3.1 Positive Carathéodory interpolation on the polydisk
    2. 3.2 Inverses of multivariable Toeplitz matrices and Christoffel-Darboux formulas
    3. 3.3 Two-variable moment problem for Bernstein-Szegő measures
    4. 3.4 Fejér-Riesz factorization and sums of Hermitian squares
    5. 3.5 Completion problems for positive semidefinite functions on amenable groups
    6. 3.6 Moment problems on free groups
    7. 3.7 Noncommutative factorization
    8. 3.8 Two-variable Hamburger moment problem
    9. 3.9 Bochner’s theorem and an application to autoregressive stochastic processes
    10. 3.10 Exercises
    11. 3.11 Notes
  11. Chapter 4. Contractive analogs
    1. 4.1 Contractive operator-matrix completions
    2. 4.2 Linearly constrained completion problems
    3. 4.3 The operator-valued Nehari and Carathéodory problems
    4. 4.4 Nehari’s problem in two variables
    5. 4.5 Nehari and Carathéodory problems for functions on compact groups
    6. 4.6 The Nevanlinna-Pick problem
    7. 4.7 The operator Corona problem
    8. 4.8 Joint operator/Hilbert-Schmidt norm control extensions
    9. 4.9 An L∞ extension problem for polynomials
    10. 4.10 Superoptimal completions
    11. 4.11 Superoptimal approximations of analytic functions
    12. 4.12 Model matching
    13. 4.13 Exercises
    14. 4.14 Notes
  12. Chapter 5. Hermitian and related completion problems
    1. 5.1 Hermitian completions
    2. 5.2 Ranks of completions
    3. 5.3 Minimal negative and positive signature
    4. 5.4 Inertia of Hermitian matrix expressions
    5. 5.5 Bounds for eigenvalues of Hermitian completions
    6. 5.6 Bounds for singular values of completions of partial triangular matrices
    7. 5.7 Moment problems for real measures on the unit circle
    8. 5.8 Euclidean distance matrix completions
    9. 5.9 Normal completions
    10. 5.10 Application to minimal representation of discrete systems
    11. 5.11 The separability problem in quantum information
    12. 5.12 Exercises
    13. 5.13 Notes
  13. Bibliography
  14. Subject Index
  15. Notation Index