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Article: Duality between some linear preserver problems. III. c-spectral norms and (skew)-symmetric matrices with fixed singular values
Title | Duality between some linear preserver problems. III. c-spectral norms and (skew)-symmetric matrices with fixed singular values |
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Authors | |
Issue Date | 1991 |
Publisher | Elsevier 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? |
Abstract | Let 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 Identifier | http://hdl.handle.net/10722/156217 |
ISSN | 2015 Impact Factor: 0.965 2015 SCImago Journal Rankings: 0.837 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | Li, CK | en_US |
dc.contributor.author | Tsing, NK | en_US |
dc.date.accessioned | 2012-08-08T08:40:52Z | - |
dc.date.available | 2012-08-08T08:40:52Z | - |
dc.date.issued | 1991 | en_US |
dc.identifier.citation | Linear Algebra And Its Applications, 1991, v. 143 C, p. 67-97 | en_US |
dc.identifier.issn | 0024-3795 | en_US |
dc.identifier.uri | http://hdl.handle.net/10722/156217 | - |
dc.description.abstract | Let 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.language | eng | en_US |
dc.publisher | Elsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/laa | en_US |
dc.relation.ispartof | Linear Algebra and Its Applications | en_US |
dc.title | Duality between some linear preserver problems. III. c-spectral norms and (skew)-symmetric matrices with fixed singular values | en_US |
dc.type | Article | en_US |
dc.identifier.email | Tsing, NK:nktsing@hku.hk | en_US |
dc.identifier.authority | Tsing, NK=rp00794 | en_US |
dc.description.nature | link_to_subscribed_fulltext | en_US |
dc.identifier.doi | 10.1016/0024-3795(91)90007-J | - |
dc.identifier.scopus | eid_2-s2.0-44949277240 | en_US |
dc.identifier.volume | 143 | en_US |
dc.identifier.issue | C | en_US |
dc.identifier.spage | 67 | en_US |
dc.identifier.epage | 97 | en_US |
dc.identifier.isi | WOS:A1991EH80300006 | - |
dc.publisher.place | United States | en_US |
dc.identifier.scopusauthorid | Li, CK=8048590800 | en_US |
dc.identifier.scopusauthorid | Tsing, NK=6602663351 | en_US |