On the computation of the restricted singular value decomposition via the cosine-sine decomposition

Abstract

In this paper, we show that the restricted singular value decomposition of a matrix triplet A ∈ Rn x m, B ∈ Rn x l, C ∈ Rp x m can be computed by means of the cosine-sine decomposition. In the first step, the matrices A, B, C are reduced to a lower-dimensional matrix triplet A, B, C, in which B and C are nonsingular, using orthogonal transformations such as the QR-factorization with column pivoting and the URV decomposition. In the second step, the components of the restricted singular value decomposition of A, B, C are derived from the singular value decomposition of B-1 AC-1. Instead of explicitly forming the latter product, a link with the cosine-sine decomposition, which can be computed by Van Loan's method, is exploited. Some numerical examples are given to show the performance of the presented method.