Convexity and concavity of eigenvalue sums

Elliott H. Lieb, Heinz Siedentop

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


It is well known that σ(H), the sum of the negative eigenvalues of a Hermitian matrix H, is a concave and increasing function of H. In contrast to this, we prove that for A nonsingular Hermitian and P positive definite, the function P{mapping}σ(AP)=σ(P1/2AP1/2) is convex and decreasing. Several other results of this nature are also proved.

Original languageEnglish (US)
Pages (from-to)811-816
Number of pages6
JournalJournal of Statistical Physics
Issue number5-6
StatePublished - Jun 1991

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Convexity
  • concavity
  • eigenvalues


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