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

In this paper, we investigate the singular value decomposition (SVD) of E + xyH, where E is an m x n real diagonal matrix, x E Cm, and y E Cn. We start by briefly revisiting an existing approach for determining the desired SVD by sequentially computing the eigen-decomposition of two separate hermitian rank-one modifications of a real diagonal matrix. Then we introduce the notion of the rank-two secular function whose roots are the singular values of E + 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 E + xyH in O(min(m, n ) 2 ) time.& COPY; 2023 Elsevier Inc. All rights reserved.

Automated summary

In this paper, the authors investigate the singular value decomposition (SVD) of E + xyH, where E is an m x n real diagonal matrix, x is in Cm, and y is in Cn. They propose a new method for computing the SVD of E + xyH by sequentially computing the eigen-decomposition of two separate hermitian rank-one modifications of a real diagonal matrix and exploiting the properties of the rank-two secular function. The authors also demonstrate how to compute the full set of associated left/right singular vectors in O(min(m, n)2) time.