{"title":"Comprehensive classification of the algebra generated by two idempotent matrices","authors":"Rounak Biswas, Falguni Roy","doi":"10.1016/j.laa.2024.11.005","DOIUrl":null,"url":null,"abstract":"<div><div>For two idempotent matrix <span><math><mi>P</mi><mo>,</mo><mi>Q</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span>, let alg<span><math><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span> denote the smallest subalgebra of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span> that contains <span><math><mi>P</mi><mo>,</mo><mi>Q</mi></math></span> and the identity matrix <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. This paper provides a complete classification of alg<span><math><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span> without imposing any restrictions on <em>P</em> and <em>Q</em>. As a result of this classification, the issue of group invertibility within alg<span><math><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span> is fully resolved.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 185-206"},"PeriodicalIF":1.0000,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S002437952400418X","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
For two idempotent matrix , let alg denote the smallest subalgebra of that contains and the identity matrix . This paper provides a complete classification of alg without imposing any restrictions on P and Q. As a result of this classification, the issue of group invertibility within alg is fully resolved.
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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