{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/characterizing-gsvd-by-singular-value","title":"Characterizing GSVD by singular value expansion of linear operators and its computation","arxiv_id":"2404.00655","date":"2024-03-31","proceeding":null,"authors":["Haibo Li"],"abstract":"The generalized singular value decomposition (GSVD) of a matrix pair $\\{A, L\\}$ with $A\\in\\mathbb{R}^{m\\times n}$ and $L\\in\\mathbb{R}^{p\\times n}$ 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 $\\mathcal{A}$ and $\\mathcal{L}$ induced by $\\{A, L\\}$ 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 $\\{A, L\\}$ is nothing but the SVEs of $\\mathcal{A}$ and $\\mathcal{L}$. 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 $\\{A, L\\}$, which is named the gGKB\\_GSVD algorithm. We use the GSVD of $\\{A, L\\}$ 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.","url_abs":"https://arxiv.org/abs/2404.00655v1","url_pdf":"https://arxiv.org/pdf/2404.00655v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"characterizing-gsvd-by-singular-value","repo_url":"https://github.com/machealb/gsvd_iter","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}