{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:48:19Z","timestamp":1760147299794,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,25]],"date-time":"2023-01-25T00:00:00Z","timestamp":1674604800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11971294","KJ2021A1198","KJ2021ZD0143","1224036"],"award-info":[{"award-number":["11971294","KJ2021A1198","KJ2021ZD0143","1224036"]}]},{"name":"College Natural Scientific Research Projects organized by Anhui Provincial Department of Education","award":["11971294","KJ2021A1198","KJ2021ZD0143","1224036"],"award-info":[{"award-number":["11971294","KJ2021A1198","KJ2021ZD0143","1224036"]}]},{"name":"Beijing Natural Science Foundation","award":["11971294","KJ2021A1198","KJ2021ZD0143","1224036"],"award-info":[{"award-number":["11971294","KJ2021A1198","KJ2021ZD0143","1224036"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We address classification of permutation matrices, in terms of permutation similarity relations, which play an important role in investigating the reducible solutions of some symmetric matrix equations. We solve the three problems. First, what is the canonical form of a permutation similarity class? Second, how to obtain the standard form of arbitrary permutation matrix? Third, for any permutation matrix A, how to find the permutation matrix T, such that T\u22121AT is in canonical form? Besides, the decomposition theorem of permutation matrices and the factorization theorem of both permutation matrices and monomial matrices are demonstrated.<\/jats:p>","DOI":"10.3390\/sym15020332","type":"journal-article","created":{"date-parts":[[2023,1,25]],"date-time":"2023-01-25T03:53:18Z","timestamp":1674618798000},"page":"332","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Canonical Forms of Permutation Matrices"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0271-4491","authenticated-orcid":false,"given":"Wen-Wei","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematical Science, University of Science and Technology of China, Hefei 230026, China"},{"name":"School of Information and Mathematics, Anhui International Studies University, Hefei 231201, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3991-2344","authenticated-orcid":false,"given":"Xin","family":"Hou","sequence":"additional","affiliation":[{"name":"College of Elementary Education, Capital Normal University, Beijing 100048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0189-5355","authenticated-orcid":false,"given":"Qing-Wen","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai University, Shanghai 200444, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/0012-365X(91)90280-F","article-title":"A computer search for finite projective planes of order 9","volume":"92","author":"Lam","year":"1991","journal-title":"Discret. 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