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# Matrix Completions, Moments, and Sums of Hermitian SquaresMihály Bakonyi & Hugo J. Woerdeman

 TABLE OF CONTENTS: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 170Chapter 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 250Chapter 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 352Chapter 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 470Bibliography 475Subject Index 513Notation Index 517Return to Book DescriptionFile created: 4/21/2017 Questions and comments to: webmaster@press.princeton.eduPrinceton University Press