Characterizing GSVD by Singular Value Expansion of Linear Operators and Its Computation
Haibo Li · SIAM Journal on Matrix Analysis and Applications · 2025
Abstract. The generalized singular value decomposition (GSVD) of a matrix pair [Formula: see text] with [Formula: see text] and [Formula: see text] generalizes the singular value decomposition (SVD) of a single matrix. In this paper, we provide a new understanding of GSVD from the viewpoint of SVD, based on which we propose a new iterative method for computing nontrivial GSVD components of a large-scale matrix pair. By introducing two linear operators [Formula: see text] and [Formula: see text] induced by [Formula: see text] between two finite-dimensional Hilbert spaces and applying the theory of singular value expansion (SVE) for linear compact operators, we show that the GSVD of [Formula: see text] is nothing but the SVEs of [Formula: see text] and [Formula: see text]. This result characterizes completely the structure of GSVD for any matrix pair with the same number of columns. As a direct application of this result, we generalize the standard Golub–Kahan bidiagonalization (GKB) that is a basic routine for large-scale SVD computation such that the resulting generalized GKB (gGKB) process can be used to approximate nontrivial extreme GSVD components of [Formula: see text], which is named the gGKB_GSVD algorithm. We use the GSVD of [Formula: see text] to study several basic properties of gGKB and also provide preliminary results about convergence and accuracy of gGKB_GSVD for GSVD computation. Numerical experiments are presented to demonstrate the effectiveness of this method.