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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 infinite-length 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
|Number of pages||14|
|Journal||IEEE Transactions on Signal Processing|
|Early online date||6 Mar 2018|
|Publication status||Published - 15 May 2018|
- hermitian matrix
- parahermitian matrix
- eigenvalue decomposition algorithm
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- Electronic And Electrical Engineering - Professor
- 1 Finished
Signal Processing Solutions for the Networked Battlespace
EPSRC (Engineering and Physical Sciences Research Council)
1/04/13 → 31/03/18
Eigenvalue decomposition of a parahermitian matrix: extraction of analytic EigenvectorsWeiss, S., Proudler, I., Coutts, F. K. & Khattak, F. A., 24 Apr 2023, (E-pub ahead of print) In: IEEE Transactions on Signal Processing. 71, p. 1642-1656 15 p.
Research output: Contribution to journal › Article › peer-reviewOpen AccessFile5 Downloads (Pure)
Polynomial eigenvalue decomposition for multichannel broadband signal processingNeo, V. W., Redif, S., McWhirter, J. G., Pestana, J., Proudler, I. K., Weiss, S. & Naylor, P. A., 4 Apr 2023, (Accepted/In press) In: IEEE Signal Processing Magazine. 18 p.
Research output: Contribution to journal › Article › peer-reviewOpen AccessFile
Space-time covariance matrix estimation: loss of algebraic multiplicities of EigenvaluesKhattak, F. A., Weiss, S., Proudler, I. K. & McWhirter, J. G., 3 Nov 2022, p. 975-979. 5 p.
Research output: Contribution to conference › Paper › peer-reviewOpen AccessFile