{"title":"Projectively and Weakly Simultaneously Diagonalizable Matrices and their Applications","authors":"Wentao Ding, Jianze Li, Shuzhong Zhang","doi":"10.1137/22m1507656","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 167-202, March 2024. <br/> Abstract. Characterizing simultaneously diagonalizable (SD) matrices has been receiving considerable attention in recent decades due to its wide applications and its role in matrix analysis. However, the notion of SD matrices is arguably still restrictive for wider applications. In this paper, we consider two error measures related to the simultaneous diagonalization of matrices and propose several new variants of SD thereof; in particular, TWSD, TWSD-B, [math]-SD (SDO), DWSD, and [math]-SD (SDO). Those are all weaker forms of SD. We derive various sufficient and/or necessary conditions of them under different assumptions and show the relationships between these new notions. Finally, we discuss the applications of these new notions in, e.g., quadratically constrained quadratic programming and independent component analysis.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5000,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Matrix Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/22m1507656","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 167-202, March 2024. Abstract. Characterizing simultaneously diagonalizable (SD) matrices has been receiving considerable attention in recent decades due to its wide applications and its role in matrix analysis. However, the notion of SD matrices is arguably still restrictive for wider applications. In this paper, we consider two error measures related to the simultaneous diagonalization of matrices and propose several new variants of SD thereof; in particular, TWSD, TWSD-B, [math]-SD (SDO), DWSD, and [math]-SD (SDO). Those are all weaker forms of SD. We derive various sufficient and/or necessary conditions of them under different assumptions and show the relationships between these new notions. Finally, we discuss the applications of these new notions in, e.g., quadratically constrained quadratic programming and independent component analysis.
期刊介绍：
The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.
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