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Furthermore, this paper addresses the inverse problem, that is, given a bisymmetric, doubly quasi-stochastic tridiagonal Jacobi matrix U of order n, determine under what conditions there exists a real tridiagonal Toeplitz matrix A such that its combined matrix is U.<\/jats:p>","DOI":"10.3390\/axioms14050375","type":"journal-article","created":{"date-parts":[[2025,5,16]],"date-time":"2025-05-16T06:57:28Z","timestamp":1747378648000},"page":"375","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Combined Matrix of a Tridiagonal Toeplitz Matrix"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9837-3926","authenticated-orcid":false,"given":"Bego\u00f1a","family":"Cant\u00f3","sequence":"first","affiliation":[{"name":"Instituto de Matem\u00e1tica Multidisciplinar, Universitat Polit\u00e8cnica de Val\u00e8ncia, 46071 Val\u00e8ncia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1341-2800","authenticated-orcid":false,"given":"Rafael","family":"Cant\u00f3","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica Multidisciplinar, Universitat Polit\u00e8cnica de Val\u00e8ncia, 46071 Val\u00e8ncia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8590-1243","authenticated-orcid":false,"given":"Ana Maria","family":"Urbano","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica Multidisciplinar, Universitat Polit\u00e8cnica de Val\u00e8ncia, 46071 Val\u00e8ncia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1016\/j.laa.2017.06.010","article-title":"Explicit inverse of a tridiagonal (p,r)-Toeplitz matrix","volume":"542","author":"Encinas","year":"2018","journal-title":"Linear Algebra Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/s40314-020-01369-3","article-title":"Fast solvers for tridiagonal Toeplitz linear systems","volume":"39","author":"Liu","year":"2020","journal-title":"Comput. 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