About: Golden–Thompson inequality

An Entity of Type: Thing, from Named Graph: http://dbpedia.org, within Data Space: dbpedia.org

Trace inequality between traces of exponential of symmetric matrices

Property Value
dbo:description
  • mathematischer Satz (de)
  • trace inequality between traces of exponential of symmetric matrices (en)
dbo:wikiPageExternalLink
dbo:wikiPageWikiLink
dbp:authorlink
  • Bertram Kostant (en)
dbp:border
  • 1 (xsd:integer)
dbp:first
  • Bertram (en)
dbp:last
  • Kostant (en)
dbp:mathStatement
  • Given Hermitian , if then . (en)
  • Given Hermitian matrices , (en)
  • If are Hermitian and positive semidefinite, then (en)
  • For any integer , . For any integer , . At limit, we obtain the operator norm . (en)
dbp:name
  • Theorem (en)
  • Golden–Thompson inequality (en)
  • Corollary (en)
  • Golden inequality (en)
dbp:proof
  • By the Lie product formula, . By the Golden inequality, . (en)
  • For any , we have , thus . Thus is a contraction map, thus , thus , thus all eigenvalues of are nonpositive, thus . (en)
  • If for all , then all the other inequalities are also proven as special cases of it. So it suffices to prove that inequality. case is trivial. case. Since are Hermitian and PSD, we can split to , which allows us to write , meaning it is a non-negative real number. Now by Cauchy–Schwarz inequality, case. Define two sequences of matrices which, by construction, are Hermitian and positive semidefinite. For any , by the cyclic property of trace, By the same argument as case, . Apply Cauchy–Schwarz, and the cyclic equalities, If , then . Otherwise, by induction, and continuing the same argument,. This continues until we obtain . (en)
  • It suffices to show that . by the Golden inequality. The second claim is proven similarly. (en)
dbp:style
  • width:100% (en)
dbp:ta
  • center (en)
dbp:title
  • Proof (en)
dbp:wikiPageUsesTemplate
dbp:year
  • 1973 (xsd:integer)
dct:subject
rdfs:label
  • Golden–Thompson inequality (en)
  • 골든-톰슨 부등식 (ko)
  • 高登─湯普森不等式 (zh)
owl:sameAs
prov:wasDerivedFrom
foaf:isPrimaryTopicOf
is dbo:wikiPageRedirects of
is dbo:wikiPageWikiLink of
is foaf:primaryTopic of
Powered by OpenLink Virtuoso    This material is Open Knowledge     W3C Semantic Web Technology     This material is Open Knowledge    Valid XHTML + RDFa
This content was extracted from Wikipedia and is licensed under the Creative Commons Attribution-ShareAlike 4.0 International