{"title":"On the characterization of generalized inverses by bordered matrices","authors":"Kentaro Nomakuchi","doi":"10.1016/0024-3795(80)90093-2","DOIUrl":null,"url":null,"abstract":"<div><p>A method to characterize the class of all generalized inverses of any given matrix <em>A</em> is considered. Given a matrix <em>A</em> and a nonsingular bordered matrix <em>T</em> of <em>A</em>, <span><span><span><math><mtext>T=</mtext><mtext>A</mtext><mtext>P</mtext><mtext>Q</mtext><mtext>R</mtext></math></span></span></span> the submatrix, corresponding to <em>A</em>, of <em>T</em><sup>-1</sup> is a generalized inverse of <em>A</em>, and conversely, any generalized inverse of <em>A</em> is obtainable by this method. There are different definitions of a generalized inverse, and the arguments are developed with the least restrictive definition. The characterization of the Moore-Penrose inverse, the most restrictive definition, is also considered.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"33 ","pages":"Pages 1-8"},"PeriodicalIF":1.0000,"publicationDate":"1980-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/0024-3795(80)90093-2","citationCount":"11","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/0024379580900932","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: 11
Abstract
A method to characterize the class of all generalized inverses of any given matrix A is considered. Given a matrix A and a nonsingular bordered matrix T of A, the submatrix, corresponding to A, of T-1 is a generalized inverse of A, and conversely, any generalized inverse of A is obtainable by this method. There are different definitions of a generalized inverse, and the arguments are developed with the least restrictive definition. The characterization of the Moore-Penrose inverse, the most restrictive definition, is also considered.
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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