{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T03:02:51Z","timestamp":1760151771966,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,28]],"date-time":"2022-11-28T00:00:00Z","timestamp":1669593600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12161020","12061025","Qian-Jiao-He KY Zi [2020]298"],"award-info":[{"award-number":["12161020","12061025","Qian-Jiao-He KY Zi [2020]298"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Educational Commission of Guizhou Province","award":["12161020","12061025","Qian-Jiao-He KY Zi [2020]298"],"award-info":[{"award-number":["12161020","12061025","Qian-Jiao-He KY Zi [2020]298"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Given a complex idempotent matrix A, we derive simple, sufficient and necessary conditions for a matrix X being a nontrivial solution of the Yang-Baxter-like matrix equation AXA\u00a0=\u00a0XAX, discriminating commuting solutions from non-commuting ones. On this basis, we construct all the commuting solutions of the nonlinear matrix equation.<\/jats:p>","DOI":"10.3390\/sym14122510","type":"journal-article","created":{"date-parts":[[2022,11,28]],"date-time":"2022-11-28T04:01:33Z","timestamp":1669608093000},"page":"2510","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Solution Set of the Yang-Baxter-like Matrix Equation for an Idempotent Matrix"],"prefix":"10.3390","volume":"14","author":[{"given":"Xiaoling","family":"Xu","sequence":"first","affiliation":[{"name":"College of Mathematics and Data Science (Software College), Minjiang University, Fuzhou 350108, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Linzhang","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, China"},{"name":"School of Mathematical Sciences, Xiamen University, Xiamen 361005, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6593-7169","authenticated-orcid":false,"given":"Qilong","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1137\/S0895479801397275","article-title":"Solution form and simple iteration of a nonsymmetric algebraic Riccati equation arising in transport theory","volume":"26","author":"Lu","year":"2005","journal-title":"SIAM J. 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