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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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Hermitian and related completion problems

Hermitian and related completion problems

(p.361) Chapter Five Hermitian and related completion problems
Matrix Completions, Moments, and Sums of Hermitian Squares

Mihály Bakonyi

Hugo J. Woerdeman

Princeton University Press

This chapter considers various completion problems that are in one way or another closely related to positive semidefinite or contractive completion problems. For instance, as a variation on requiring that all eigenvalues of the completion are positive/nonnegative, one can consider the question how many eigenvalues of a Hermitian completion have to be positive/nonnegative. In the solution to the latter problem ranks of off-diagonal parts will play a role, which is why minimal rank completions are also discussed. Related is a question on real measures on the real line. As a variation of the contractive completion problem, the chapter considers the question how many singular values of a completion have to be smaller (or larger) than one. It also looks at completions in classes of normal matrices and distance matrices. As applications it turns to questions regarding Hermitian matrix expressions, a minimal representation problem for discrete systems, and the separability problem that appears in quantum information. Exercises and notes are provided at the end of the chapter.

Keywords:   completion problems, positive semidefinite completion, contractive completion, minimal rank completions, Hermitian matrix expressions, separability problem

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