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Article: Duality between some linear preserver problems. III. c-spectral norms and (skew)-symmetric matrices with fixed singular values

TitleDuality between some linear preserver problems. III. c-spectral norms and (skew)-symmetric matrices with fixed singular values
Authors
Issue Date1991
PublisherElsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/laa
Citation
Linear Algebra And Its Applications, 1991, v. 143 C, p. 67-97 How to Cite?
AbstractLet F denote either the complex field C or the real field R. Let V be Sn(F) or Kn(F), the vector spaces of all n × n symmetric and skew-symmetric matrices, respectively, over F. For c=(c1,...,cn)≠0 with c1≥ ⋯ ≥cn≥0, the c-spectral norm of a matrix A∈V is the quantity {norm of matrix}A{norm of matrix}c = ∑ i=l nciσi(A), where σ1(A)≥ ⋯ ≥σn(A) are the singular values of A. Let d=(d1,...,dn)≠0 with d1≥ ⋯ ≥dn≥0. We study the linear isometries between the normed spaces (V,{norm of matrix}·{norm of matrix}c) and (V,{norm of matrix}·{norm of matrix}d), by using the fact that they are dual transformations of the linear operators which map ∑(d) onto ∑(c), where ∑(c) = {X∈V:X has singular values c1,...,cn}. It is shown that such isometries (and hence their dual transformations) exist if and only if c and d are scalar multiples of each other. In such case, we completely determine the structure of such isometries, and prove that they and their dual transformations belong to a same class of operators. In the proof, we obtain characterizations of the extreme points of the unit ball in V (for different cases) with respect to {norm of matrix}·{norm of matrix}c, which is of independent interest. © 1991.
Persistent Identifierhttp://hdl.handle.net/10722/156217
ISSN
2015 Impact Factor: 0.965
2015 SCImago Journal Rankings: 0.837
ISI Accession Number ID

 

DC FieldValueLanguage
dc.contributor.authorLi, CKen_US
dc.contributor.authorTsing, NKen_US
dc.date.accessioned2012-08-08T08:40:52Z-
dc.date.available2012-08-08T08:40:52Z-
dc.date.issued1991en_US
dc.identifier.citationLinear Algebra And Its Applications, 1991, v. 143 C, p. 67-97en_US
dc.identifier.issn0024-3795en_US
dc.identifier.urihttp://hdl.handle.net/10722/156217-
dc.description.abstractLet F denote either the complex field C or the real field R. Let V be Sn(F) or Kn(F), the vector spaces of all n × n symmetric and skew-symmetric matrices, respectively, over F. For c=(c1,...,cn)≠0 with c1≥ ⋯ ≥cn≥0, the c-spectral norm of a matrix A∈V is the quantity {norm of matrix}A{norm of matrix}c = ∑ i=l nciσi(A), where σ1(A)≥ ⋯ ≥σn(A) are the singular values of A. Let d=(d1,...,dn)≠0 with d1≥ ⋯ ≥dn≥0. We study the linear isometries between the normed spaces (V,{norm of matrix}·{norm of matrix}c) and (V,{norm of matrix}·{norm of matrix}d), by using the fact that they are dual transformations of the linear operators which map ∑(d) onto ∑(c), where ∑(c) = {X∈V:X has singular values c1,...,cn}. It is shown that such isometries (and hence their dual transformations) exist if and only if c and d are scalar multiples of each other. In such case, we completely determine the structure of such isometries, and prove that they and their dual transformations belong to a same class of operators. In the proof, we obtain characterizations of the extreme points of the unit ball in V (for different cases) with respect to {norm of matrix}·{norm of matrix}c, which is of independent interest. © 1991.en_US
dc.languageengen_US
dc.publisherElsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/laaen_US
dc.relation.ispartofLinear Algebra and Its Applicationsen_US
dc.titleDuality between some linear preserver problems. III. c-spectral norms and (skew)-symmetric matrices with fixed singular valuesen_US
dc.typeArticleen_US
dc.identifier.emailTsing, NK:nktsing@hku.hken_US
dc.identifier.authorityTsing, NK=rp00794en_US
dc.description.naturelink_to_subscribed_fulltexten_US
dc.identifier.doi10.1016/0024-3795(91)90007-J-
dc.identifier.scopuseid_2-s2.0-44949277240en_US
dc.identifier.volume143en_US
dc.identifier.issueCen_US
dc.identifier.spage67en_US
dc.identifier.epage97en_US
dc.identifier.isiWOS:A1991EH80300006-
dc.publisher.placeUnited Statesen_US
dc.identifier.scopusauthoridLi, CK=8048590800en_US
dc.identifier.scopusauthoridTsing, NK=6602663351en_US

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