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

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.