{"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/geometry-of-the-symplectic-stiefel-manifold","title":"Geometry of the symplectic Stiefel manifold endowed with the Euclidean metric","arxiv_id":"2103.00459","date":"2021-02-28","proceeding":null,"authors":["Bin Gao","Nguyen Thanh Son","P. -A. Absil","Tatjana Stykel"],"abstract":"The symplectic Stiefel manifold, denoted by $\\mathrm{Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $\\mathbb{R}^{2p}$ and $\\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\\times 2n$ symplectic matrices. We study the Riemannian geometry of this manifold viewed as a Riemannian submanifold of the Euclidean space $\\mathbb{R}^{2n\\times 2p}$. The corresponding normal space and projections onto the tangent and normal spaces are investigated. Moreover, we consider optimization problems on the symplectic Stiefel manifold. We obtain the expression of the Riemannian gradient with respect to the Euclidean metric, which then used in optimization algorithms. Numerical experiments on the nearest symplectic matrix problem and the symplectic eigenvalue problem illustrate the effectiveness of Euclidean-based algorithms.","url_abs":"https://arxiv.org/abs/2103.00459v1","url_pdf":"https://arxiv.org/pdf/2103.00459v1.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":"geometry-of-the-symplectic-stiefel-manifold","repo_url":"https://github.com/opt-gaobin/spopt","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}