{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,24]],"date-time":"2025-10-24T08:32:27Z","timestamp":1761294747618,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,18]],"date-time":"2024-06-18T00:00:00Z","timestamp":1718668800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004488","name":"Croatian Science Foundation","doi-asserted-by":"publisher","award":["IP-2020-02-2240"],"award-info":[{"award-number":["IP-2020-02-2240"]}],"id":[{"id":"10.13039\/501100004488","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This article considers arrowhead and diagonal-plus-rank-one matrices in Fn\u00d7n where F\u2208{R,C,H} and where H is a noncommutative algebra of quaternions. We provide unified formulas for fast determinants and inverses for considered matrices. The formulas are unified in the sense that the same formula holds in both commutative and noncommutative associative fields or algebras, with noncommutative examples being matrices of quaternions and block matrices. Each formula requires O(n) arithmetic operations, as does multiplication of such matrices with a vector. The formulas are efficiently implemented using the polymorphism or multiple-dispatch feature of the Julia programming language.<\/jats:p>","DOI":"10.3390\/axioms13060409","type":"journal-article","created":{"date-parts":[[2024,6,18]],"date-time":"2024-06-18T03:42:06Z","timestamp":1718682126000},"page":"409","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Inverses and Determinants of Arrowhead and Diagonal-Plus-Rank-One Matrices over Associative Algebras"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9037-7631","authenticated-orcid":false,"given":"Nevena","family":"Jakov\u010devi\u0107 Stor","sequence":"first","affiliation":[{"name":"Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Ru\u0111era Bo\u0161kovi\u0107a 32, 21000 Split, Croatia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8741-3988","authenticated-orcid":false,"given":"Ivan","family":"Slapni\u010dar","sequence":"additional","affiliation":[{"name":"Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Ru\u0111era Bo\u0161kovi\u0107a 32, 21000 Split, Croatia"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,18]]},"reference":[{"key":"ref_1","unstructured":"Golub, G.H., and Loan, C.F.V. (2013). Matrix Computations, The John Hopkins University Press. [4th ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.aml.2014.02.010","article-title":"An efficient method for computing the inverse of arrowhead matrices","volume":"33","author":"Najafi","year":"2014","journal-title":"Appl. Math. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.laa.2013.10.007","article-title":"Accurate eigenvalue decomposition of real symmetric arrowhead matrices and applications","volume":"464","author":"Stor","year":"2015","journal-title":"Linear Algebra Appl."},{"key":"ref_4","unstructured":"Hamilton, W.R. (1853). Lectures on Quaternions, Hodges and Smith."},{"key":"ref_5","unstructured":"Hamilton, W.E. (1866). Elements of Quaternions, Longmans, Green, and Co."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1017\/S0305004100055638","article-title":"Quaternionic analysis","volume":"85","author":"Sudbery","year":"1979","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_7","unstructured":"(2024, May 02). The Julia Language. Available online: http:\/\/julialang.org\/."},{"key":"ref_8","unstructured":"(2024, May 02). Quaternions.jl. Available online: https:\/\/github.com\/JuliaGeometry\/Quaternions.jl."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Chaysri, T., Stor, N.J., and Slapni\u010dar, I. (2024). Fast Eigenvalue Decomposition of Arrowhead and Diagonal-Plus-Rank-k Matrices of Quaternions. Mathematics, 12.","DOI":"10.3390\/math12091327"},{"key":"ref_10","unstructured":"Bernstein, D.S. (2009). Matrix Mathematics, Princeton University Press."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/j.laa.2015.09.025","article-title":"Forward stable eigenvalue decomposition of rank-one modifications of diagonal matrices","volume":"487","author":"Stor","year":"2015","journal-title":"Linear Algebra Appl."},{"key":"ref_12","unstructured":"Powell, P.D. (2011). Calculating determinants of block matrices. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/BF03024312","article-title":"Quaternionic determinants","volume":"18","author":"Aslaksen","year":"1996","journal-title":"Math. Intell."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"100","DOI":"10.13001\/1081-3810.1050","article-title":"The quaternionic determinant","volume":"7","author":"Cohen","year":"2000","journal-title":"Electron. J. Linear Algebra"},{"key":"ref_15","unstructured":"(2024, May 02). Matrix Algorithms in Noncommutative Associative Algebras. Available online: https:\/\/github.com\/ivanslapnicar\/MANAA\/."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/409\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:00:19Z","timestamp":1760108419000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/409"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,18]]},"references-count":15,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["axioms13060409"],"URL":"https:\/\/doi.org\/10.3390\/axioms13060409","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,6,18]]}}}