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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.

Mathematics --- Matrices. --- Bernstein–Szeg ő measures. --- Carathéodory problem. --- Christoel–Darboux formulas. --- Corona problem. --- Fejér–Riesz factorization. --- Hamburger problem. --- Hermitian matrices. --- Hermitian matrix expressions. --- Hermitian squares problems. --- Hilbert spaces. --- Hilbert–Schmidt norm control. --- MATLAB codes. --- Nehari problem. --- Nevanlinna–Pick problem. --- Schur complement. --- Toeplitz case. --- Toeplitz matrices. --- banded case. --- chordal case. --- completion problems. --- complex function theory. --- cones. --- contractive completions. --- control theory. --- electrical engineering. --- linear algebra. --- mathematics. --- measure theory. --- minimal rank completions. --- multivariables. --- operator theory. --- partial operator matrices. --- positive Carathéodory interpolation. --- positive definite completions. --- positive semidefinite completion. --- quantum information. --- semidefinite completions. --- semidefinite matrices. --- semidefinite operator matrices. --- semidefinite programming. --- separability problem. --- signal processing. --- trigonometric polynomials.