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Matrix Completions, Moments, and Sums of Hermitian Squares$
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Mihály Bakonyi and Hugo J. Woerdeman

Print publication date: 2011

Print ISBN-13: 9780691128894

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691128894.001.0001

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Completions of positive semidefinite operator matrices

Completions of positive semidefinite operator matrices

Chapter:
(p.69) Chapter Two Completions of positive semidefinite operator matrices
Source:
Matrix Completions, Moments, and Sums of Hermitian Squares
Author(s):

Mihály Bakonyi

Hugo J. Woerdeman

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691128894.003.0002

This chapter deals with positive definite and semidefinite completions of partial operator matrices. It considers the banded case in Section 2.1, the chordal case in Section 2.2, the Toeplitz case in Section 2.3, and the generalized banded case and the operator-valued positive semidefinite chordal case in Section 2.6. Section 2.4 introduces the Schur complement and uses it to derive an operator-valued Fejér–Riesz factorization. Section 2.5 is devoted to describing the structure of positive semidefinite operator matrices. Section 2.7 studies the Hamburger problem based on positive semidefinite completions of Hankel matrices. Finally, Section 2.8 indicates the connection with linear prediction. Exercises and notes are provided at the end of the chapter.

Keywords:   positive definite completions, semidefinite completions, partial operator matrices, banded case, chordal case, Toeplitz case, Schur complement, Fejér–Riesz factorization, semidefinite operator matrices, Hamburger problem

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