A Generalized Schur–Horn Theorem and Optimal Frame Completions

Authors

Matthew C. Fickus, Air Force Institute of TechnologyFollow
Justin D. Marks, Bowdoin College
Miriam J. Poteet, Air Force Institute of Technology

Document Type

Article

Publication Date

5-2016

Abstract

The Schur-Horn theorem is a classical result in matrix analysis which characterizes the existence of positive semi-definite matrices with a given diagonal and spectrum. In recent years, this theorem has been used to characterize the existence of finite frames whose elements have given lengths and whose frame operator has a given spectrum. We provide a new generalization of the Schur-Horn theorem which characterizes the spectra of all possible finite frame completions. That is, we characterize the spectra of the frame operators of the finite frames obtained by adding new vectors of given lengths to an existing frame. We then exploit this characterization to give a new and simple algorithm for computing the optimal such completion.

Comments

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 This record previously pointed at the arXiv.org preprint of the article at arXiv:1408.2882 [math.FA]

DOI

10.1016/j.acha.2015.03.004

Source Publication

Applied and Computational Harmonic Analysis (ISSN 1063-5203 | eISSN 1096-603X)

Recommended Citation

Fickus, M. C., Marks, J. D., & Poteet, M. J. (2016). A generalized Schur–Horn theorem and optimal frame completions. Applied and Computational Harmonic Analysis, 40(3), 505–528. https://doi.org/10.1016/j.acha.2015.03.004