{"title":"On the singular values of a product of matrices","authors":"W. Watkins","doi":"10.6028/JRES.074B.025","DOIUrl":null,"url":null,"abstract":"The purpose of this note is to give necessary and sufficient conditions for the singular values of a product of matrices to be equal to certain products of their singular values. We then analyze the case of equa]jty in a matrix inequality of Os trows ki . Th e s ingular values of an n·square co mplex matrix X are the positive square roots of the eigen· values of X*X, where X* is the conjugate tran spose of X. Denote the singular values of X by a t (X) , ... , an (X), arranged so that al (X) ~ ... ~ an(X) > 0 (all matrices are assumed to be nonsi ngular). Let A and B be n,·square complex matri ces and let A = UH, B = VK be the polar factorizations of A a nd B. In the factorization s U and V are unitary matrices and Hand K are positive·definit e hermitian matri ces. THEOREM 1: Let k be a positive integer less than n. Then","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"69 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.6028/JRES.074B.025","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
Abstract
The purpose of this note is to give necessary and sufficient conditions for the singular values of a product of matrices to be equal to certain products of their singular values. We then analyze the case of equa]jty in a matrix inequality of Os trows ki . Th e s ingular values of an n·square co mplex matrix X are the positive square roots of the eigen· values of X*X, where X* is the conjugate tran spose of X. Denote the singular values of X by a t (X) , ... , an (X), arranged so that al (X) ~ ... ~ an(X) > 0 (all matrices are assumed to be nonsi ngular). Let A and B be n,·square complex matri ces and let A = UH, B = VK be the polar factorizations of A a nd B. In the factorization s U and V are unitary matrices and Hand K are positive·definit e hermitian matri ces. THEOREM 1: Let k be a positive integer less than n. Then