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.

Preface ixChapter 1. Cones of Hermitian matrices and trigonometric polynomials 11.1 Cones and their basic properties 11.2 Cones of Hermitian matrices 61.3 Cones of trigonometric polynomials 131.4 Determinant and entropy maximization 411.5 Semidefinite programming 471.6 Exercises 511.7 Notes 65Chapter 2. Completions of positive semidefinite operator matrices 692.1 Positive definite completions: the banded case 692.2 Positive definite completions: the chordal case 762.3 Positive definite completions: the Toeplitz case 832.4 The Schur complement and Fejér-Riesz factorization 982.5 Schur parameters 1152.6 The central completion, maximum entropy, and inheritance principle 1242.7 The Hamburger moment problem and spectral factorization on the real line 1382.8 Linear prediction 1562.9 Exercises 1572.10 Notes 170
Chapter 3. Multivariable moments and sums of Hermitian squares 1753.1 Positive Carathéodory interpolation on the polydisk 1763.2 Inverses of multivariable Toeplitz matrices and Christoffel-Darboux formulas 1873.3 Two-variable moment problem for Bernstein-Szegö measures 1983.4 Fejér-Riesz factorization and sums of Hermitian squares 2083.5 Completion problems for positive semidefinite functions on amenable groups 2113.6 Moment problems on free groups 2143.7 Noncommutative factorization 2233.8 Two-variable Hamburger moment problem 2283.9 Bochner's theorem and an application to autoregressive stochastic processes 2353.10 Exercises 2403.11 Notes 250
Chapter 4. Contractive analogs 2574.1 Contractive operator-matrix completions 2584.2 Linearly constrained completion problems 2694.3 The operator-valued Nehari and Carathéodory problems 2754.4 Nehari's problem in two variables 2864.5 Nehari and Carathéodory problems for functions on compact groups 2924.6 The Nevanlinna-Pick problem 2994.7 The operator Corona problem 3084.8 Joint operator/Hilbert-Schmidt norm control extensions 3144.9 An L1 extension problem for polynomials 3174.10 Superoptimal completions 3214.11 Superoptimal approximations of analytic functions 3294.12 Model matching 3394.13 Exercises 3414.14 Notes 352
Chapter 5. Hermitian and related completion problems 3615.1 Hermitian completions 3615.2 Ranks of completions 3745.3 Minimal negative and positive signature 3835.4 Inertia of Hermitian matrix expressions 4015.5 Bounds for eigenvalues of Hermitian completions 4075.6 Bounds for singular values of completions of partial triangular matrices 4135.7 Moment problems for real measures on the unit circle 4185.8 Euclidean distance matrix completions 4265.9 Normal completions 4335.10 Application to minimal representation of discrete systems 4425.11 The separability problem in quantum information 4435.12 Exercises 4515.13 Notes 470
Bibliography 475Subject Index 513Notation Index 517