{"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/block-diagonalization-of-quaternion-circulant","title":"Block Diagonalization of Quaternion Circulant Matrices with Applications","arxiv_id":"2302.04086","date":"2023-02-08","proceeding":null,"authors":["Junjun Pan","Michael K. Ng"],"abstract":"It is well-known that a complex circulant matrix can be diagonalized by a discrete Fourier matrix with imaginary unit $\\mathtt{i}$. The main aim of this paper is to demonstrate that a quaternion circulant matrix cannot be diagonalized by a discrete quaternion Fourier matrix with three imaginary units $\\mathtt{i}$, $\\mathtt{j}$ and $\\mathtt{k}$. Instead, a quaternion circulant matrix can be block-diagonalized into 1-by-1 block and 2-by-2 block matrices by permuted discrete quaternion Fourier transform matrix. With such a block-diagonalized form, the inverse of a quaternion circulant matrix can be determined efficiently similar to the inverse of a complex circulant matrix. We make use of this block-diagonalized form to study quaternion tensor singular value decomposition of quaternion tensors where the entries are quaternion numbers. The applications including computing the inverse of a quaternion circulant matrix, and solving quaternion Toeplitz system arising from linear prediction of quaternion signals are employed to validate the efficiency of our proposed block diagonalized results. A numerical example of color video as third-order quaternion tensor is employed to validate the effectiveness of quaternion tensor singular value decomposition.","url_abs":"https://arxiv.org/abs/2302.04086v2","url_pdf":"https://arxiv.org/pdf/2302.04086v2.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":"block-diagonalization-of-quaternion-circulant","repo_url":"https://github.com/panjun009/blkdiagcir_quaternion","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}