Eigenvalue decomposition of a parahermitian matrix: extraction of analytic Eigenvectors

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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 one-dimensional eigenspaces in which analytic eigenvectors can exist. Determining such eigenvectors is not straightforward, and requires three challenges to be addressed. Firstly, one-dimensional subspaces for eigenvectors have to be woven smoothly across DFT bins where a non-trivial algebraic multiplicity causes ambiguity. Secondly, with the one-dimensional 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 languageEnglish
Pages (from-to)1642-1656
Number of pages15
JournalIEEE Transactions on Signal Processing
Early online date24 Apr 2023
Publication statusE-pub ahead of print - 24 Apr 2023


  • eigenvalue decomposition
  • parahermitian matrix factorisation
  • analytic functions
  • phase retrieval
  • quadratic programming with quadratic constraints


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