Note on a rank-one modification of the singular value decomposition

James Baglama, Vasilije Perović and Timothy Toolan

Applied Mathematics and Computation, 2023, vol. 457, issue C

Abstract: In this paper, we investigate the singular value decomposition (SVD) of Σ+xyH, where Σ is an m×n real diagonal matrix, x∈Cm, and y∈Cn. We start by briefly revisiting an existing approach for determining the desired SVD by sequentially computing the eigendecomposition of two separate hermitian rank-one modifications of a real diagonal matrix. Then we introduce the notion of the rank-two secular functionwhose roots are the singular values of Σ+xyH and exploit its properties to bound each root/singular value in disjoint intervals. Once the singular values are computed, we demonstrate how to directly compute the full set of associated left/right singular vectors ultimately giving us a new method for computing the SVD of Σ+xyH in O(min(m,n)2) time.

Keywords: Singular value decomposition; Eigenvalue decomposition; Rank-one update; Rank-two secular function (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:457:y:2023:i:c:s0096300323003399

DOI: 10.1016/j.amc.2023.128170

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