{"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/operator-valued-semicircular-elements-solving","title":"Operator-valued semicircular elements: Solving a quadratic matrix equation with positivity constraints","arxiv_id":"math/0703510","date":"2007-03-17","proceeding":null,"authors":["J. William Helton","Reza Rashidi Far","Roland Speicher"],"abstract":"We show that the quadratic matrix equation $VW + \\eta (W)W = I$, for given $V$ with positive real part and given analytic mapping $\\eta$ with some positivity preserving properties, has exactly one solution $W$ with positive real part. We point out the relevance of this result in the context of operator-valued free probability theory and for the determination of the asymptotic eigenvalue distribution of band or block random matrices. We also address the problem of a numerical determination of the solution.","url_abs":"https://arxiv.org/abs/math/0703510v1","url_pdf":"https://arxiv.org/pdf/math/0703510v1.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":"operator-valued-semicircular-elements-solving","repo_url":"https://github.com/johannes-hoffmann/ncdist.jl","is_official":0,"mentioned_in_paper":0,"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}