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.
Applied and Computational Harmonic Analysis
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
This record sources the open access preprint (pre-refereed) version of the article at arXiv:1408.2882 [math.FA]; https://arxiv.org
Date of arXiv submission: 12 Aug 2014 [v1], updated 02 Apr 2015 [v2]
The publisher's version of record is a subscription-access article at ScienceDirect. The citation is noted below.