{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T07:22:53Z","timestamp":1769152973572,"version":"3.49.0"},"reference-count":27,"publisher":"Wiley","license":[{"start":{"date-parts":[[2012,1,1]],"date-time":"2012-01-01T00:00:00Z","timestamp":1325376000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11001079"],"award-info":[{"award-number":["11001079"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["yk0157124"],"award-info":[{"award-number":["yk0157124"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Excellent Young Teacher Grant of East China University of Science and Technology","award":["11001079"],"award-info":[{"award-number":["11001079"]}]},{"name":"Excellent Young Teacher Grant of East China University of Science and Technology","award":["yk0157124"],"award-info":[{"award-number":["yk0157124"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2012]]},"abstract":"<jats:p>We establish necessary and sufficient conditions for the existence of and the expressions for the general real and complex Hermitian solutions to the classical system of quaternion matrix equations<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>X<\/mml:mi><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:mi>X<\/mml:mi><mml:msub><mml:mrow><mml:mi>B<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mtext>, and\u2009\u2009<\/mml:mtext><mml:msub><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>X<\/mml:mi><mml:msubsup><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi>*<\/mml:mi><\/mml:mrow><\/mml:msubsup><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:math>. Moreover, formulas of the maximal and minimal ranks of four real matrices<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:math>, and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:math>in solution<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>X<\/mml:mi><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:msub><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>i<\/mml:mi><mml:mo>+<\/mml:mo><mml:msub><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>j<\/mml:mi><mml:mo>+<\/mml:mo><mml:msub><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>k<\/mml:mi><\/mml:math>to the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equations<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>X<\/mml:mi><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:mi>X<\/mml:mi><mml:msub><mml:mrow><mml:mi>B<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>X<\/mml:mi><mml:msubsup><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi>*<\/mml:mi><\/mml:mrow><\/mml:msubsup><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:math>, and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mtext>\u2009\u2009<\/mml:mtext><mml:msub><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>X<\/mml:mi><mml:msubsup><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi>*<\/mml:mi><\/mml:mrow><\/mml:msubsup><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:math>to have real and complex Hermitian solutions.<\/jats:p>","DOI":"10.1155\/2012\/307036","type":"journal-article","created":{"date-parts":[[2012,1,23]],"date-time":"2012-01-23T21:15:33Z","timestamp":1327353333000},"page":"1-19","source":"Crossref","is-referenced-by-count":2,"title":["The Real and Complex Hermitian Solutions to a System of Quaternion Matrix Equations with Applications"],"prefix":"10.1155","volume":"2012","author":[{"given":"Shao-Wen","family":"Yu","sequence":"first","affiliation":[{"name":"Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-6101-8"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1137\/0131050"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(94)00311-4"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1002\/nla.553"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1080\/00207160701458344"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(03)00385-9"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-010-9760-8"},{"issue":"2","key":"8","first-page":"57","volume":"21","year":"1998","journal-title":"Bulletin of the Malaysian Mathematical Society. Second Series"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2009.03.011"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479802415545"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2010.12.010"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2010.01.015"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1080\/0308108031000114631"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2006.05.011"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2007.06.023"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(02)00345-2"},{"key":"17","first-page":"345","volume":"9","year":"2003","journal-title":"New York Journal of Mathematics"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1007\/s100120200015"},{"issue":"1","key":"19","doi-asserted-by":"crossref","first-page":"40","DOI":"10.35834\/2002\/1401040","volume":"14","year":"2002","journal-title":"Missouri Journal of Mathematical Sciences"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2006.06.012"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2006.12.039"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2007.05.018"},{"key":"24","first-page":"552","volume":"20","year":"2010","journal-title":"Electronic Journal of Linear Algebra"},{"issue":"2","key":"25","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1142\/S1005386710000349","volume":"17","year":"2010","journal-title":"Algebra Colloquium"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1080\/00927870802317590"},{"key":"27","first-page":"343","volume":"17","year":"2008","journal-title":"Electronic Journal of Linear Algebra"},{"key":"28","doi-asserted-by":"publisher","DOI":"10.1080\/03081080008818664"}],"updated-by":[{"DOI":"10.1155\/2014\/151753","type":"erratum","label":"Erratum","source":"publisher","updated":{"date-parts":[[2014,9,30]],"date-time":"2014-09-30T00:00:00Z","timestamp":1412035200000}}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2012\/307036.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2012\/307036.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2012\/307036.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,7,1]],"date-time":"2020-07-01T05:23:26Z","timestamp":1593581006000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2012\/307036\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012]]},"references-count":27,"alternative-id":["307036","307036"],"URL":"https:\/\/doi.org\/10.1155\/2012\/307036","relation":{"erratum":[{"id-type":"doi","id":"10.1155\/2014\/151753","asserted-by":"object"}]},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012]]}}}