Relations between analytic spectral and singular value decompositions

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Weiss, Stephan and Barbarino, Giovanni; (2025) Relations between analytic spectral and singular value decompositions. In: 2025 33rd European Signal Processing Conference (EUSIPCO). IEEE, ITA, pp. 2272-2276. ISBN 9789464593624

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Abstract

We compare the existence and uniqueness of the analytic singular value decomposition (SVD) of a matrix A (z) to that of the analytic spectral or eigenvalue decomposition (EVD) of R (z) = A (z) A P(z). Both cases require oversampling if the matrices are connected to multiplexing operations. Additionally, the analytic SVD may require additional oversampling due to zero crossings of its singular values, which, different from ordinary matrices, cannot necessarily be constrained to be non-negative. It has recently been shown that oversampling can be compensated in parts by permitting singular values to be complex-valued holomorphic on the unit circle, and additionally for the SVD to perform a block-diagonalisation to pseudo-circulant subblocks. We demonstrate here that complex-valued singular values can also be motivated through fractional delay factors, link the SVD-variants to the analytic spectral decomposition, and discuss some aspects of the uniqueness of both the SVD and the spectral decomposition.

ORCID iDs

Weiss, Stephan ORCID logoORCID: https://orcid.org/0000-0002-3486-7206 and Barbarino, Giovanni;
CORE (COnnecting REpositories)