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Abstract
An analytic parahermitian matrix admits in almost all cases an eigenvalue decomposition (EVD) with analytic eigenvalues and eigenvectors. We have previously defined a discrete Fourier transform (DFT) domain algorithm which has been proven to extract the analytic eigenvalues. The selection of the eigenvalues as analytic functions guarantees in turn the existence of unique onedimensional eigenspaces in which analytic eigenvectors can exist. Determining such eigenvectors is not straightforward, and requires three challenges to be addressed. Firstly, onedimensional subspaces for eigenvectors have to be woven smoothly across DFT bins where a nontrivial algebraic multiplicity causes ambiguity. Secondly, with the onedimensional eigenspaces defined, a phase smoothing across DFT bins aims to extract analytic eigenvectors with minimum time domain support. Thirdly, we need to check whether the DFT length, and thus the approximation order, is sufficient. We propose an iterative algorithm for the extraction of analytic eigenvectors and prove that this algorithm converges to the best of a set of stationary points. We provide a number of numerical examples and simulation results, in which the algorithm is demonstrated to extract the ground truth analytic eigenvectors arbitrarily closely.
Original language  English 

Pages (fromto)  16421656 
Number of pages  15 
Journal  IEEE Transactions on Signal Processing 
Volume  71 
Early online date  24 Apr 2023 
DOIs  
Publication status  Epub ahead of print  24 Apr 2023 
Keywords
 eigenvalue decomposition
 parahermitian matrix factorisation
 analytic functions
 phase retrieval
 quadratic programming with quadratic constraints
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Dive into the research topics of 'Eigenvalue decomposition of a parahermitian matrix: extraction of analytic Eigenvectors'. Together they form a unique fingerprint.Projects
 1 Finished

Signal Processing in the Information Age (UDRC III)
EPSRC (Engineering and Physical Sciences Research Council)
1/07/18 → 31/03/24
Project: Research

Scalable extraction of analytic eigenvalues from a parahermitian matrix
Khattak, F. A., Proudler, I. & Weiss, S., 22 May 2024, (Accepted/In press) p. 15. 5 p.Research output: Contribution to conference › Paper › peerreview

Eigenvalue decomposition of a parahermitian matrix: extraction of analytic eigenvalues
Weiss, S., Proudler, I. K. & Coutts, F. K., 28 Feb 2021, In: IEEE Transactions on Signal Processing. 69, p. 722737 16 p.Research output: Contribution to journal › Article › peerreview
Open AccessFile23 Citations (Scopus)118 Downloads (Pure) 
Extraction of analytic eigenvectors from a parahermitian matrix
Weiss, S., Proudler, I. K., Coutts, F. K. & Deeks, J., Sept 2020. 5 p.Research output: Contribution to conference › Paper › peerreview
Open AccessFile8 Citations (Scopus)34 Downloads (Pure)