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Abstract
This paper addresses the extension of the factorisation of a Hermitian matrix by an eigenvalue decomposition (EVD) to the case of a parahermitian matrix that is analytic at least on an annulus containing the unit circle. Such parahermitian matrices contain polynomials or rational functions in the complex variable z, and arise e.g. as cross spectral density matrices in broadband array problems. Specifically, conditions for the existence and uniqueness of eigenvalues and eigenvectors of a parahermitian matrix EVD are given, such that these can be represented by a power or Laurent series that is absolutely convergent, at least on the unit circle, permitting a direct realisation in the time domain. Based on an analysis on the unit circle, we prove that eigenvalues exist as unique and convergent but likely infinitelength Laurent series. The eigenvectors can have an arbitrary phase response, and are shown to exist as convergent Laurent series if eigenvalues are selected as analytic functions on the unit circle, and if the phase response is selected such that the eigenvectors are Hölder continuous with α>½ on the unit circle. In the case of a discontinuous phase response or if spectral majorisation is enforced for intersecting eigenvalues, an absolutely convergent Laurent series solution for the eigenvectors of a parahermitian EVD does not exist. We provide some examples, comment on the approximation of a parahermitian matrix EVD by Laurent polynomial factors, and compare our findings to the solutions provided by polynomial matrix EVD
algorithms.
algorithms.
Original language  English 

Pages (fromto)  26592672 
Number of pages  14 
Journal  IEEE Transactions on Signal Processing 
Volume  66 
Issue number  10 
Early online date  6 Mar 2018 
DOIs  
Publication status  Published  15 May 2018 
Keywords
 hermitian matrix
 parahermitian matrix
 eigenvalue decomposition algorithm
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Projects
 1 Finished

Signal Processing Solutions for the Networked Battlespace
EPSRC (Engineering and Physical Sciences Research Council)
1/04/13 → 31/03/18
Project: Research

Polynomial eigenvalue decomposition for multichannel broadband signal processing: a mathematical technique offering new insights and solutions
Neo, V. W., Redif, S., McWhirter, J. G., Pestana, J., Proudler, I. K., Weiss, S. & Naylor, P. A., 8 Nov 2023, In: IEEE Signal Processing Magazine. 40, 7, p. 1837 20 p.Research output: Contribution to journal › Article › peerreview
Open AccessFile2 Citations (Scopus)122 Downloads (Pure) 
Signal compaction using polynomial EVD for spherical array processing with applications
Neo, V. W., Evers, C., Weiss, S. & Naylor, P. A., 20 Oct 2023, In: IEEE/ACM Transactions on Audio Speech and Language Processing. 31, p. 3537  3549 13 p.Research output: Contribution to journal › Article › peerreview
Open AccessFile53 Downloads (Pure) 
Eigenvalue decomposition of a parahermitian matrix: extraction of analytic Eigenvectors
Weiss, S., Proudler, I., Coutts, F. K. & Khattak, F. A., 24 Apr 2023, (Epub ahead of print) In: IEEE Transactions on Signal Processing. 71, p. 16421656 15 p.Research output: Contribution to journal › Article › peerreview
Open AccessFile7 Citations (Scopus)53 Downloads (Pure)