Convexity and concavity of eigenvalue sums

Elliott H. Lieb; Heinz Siedentop

doi:10.1007/BF01029984

Convexity and concavity of eigenvalue sums

Elliott H. Lieb, Heinz Siedentop

Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

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(P^1/2AP^1/2) is convex and decreasing. Several other results of this nature are also proved.

Original language	English (US)
Pages (from-to)	811-816
Number of pages	6
Journal	Journal of Statistical Physics
Volume	63
Issue number	5-6
DOIs	https://doi.org/10.1007/BF01029984
State	Published - Jun 1991

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

Keywords

Convexity
concavity
eigenvalues

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10.1007/BF01029984

Cite this

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title = "Convexity and concavity of eigenvalue sums",

abstract = "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.",

keywords = "Convexity, concavity, eigenvalues",

author = "Lieb, {Elliott H.} and Heinz Siedentop",

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AU - Lieb, Elliott H.

AU - Siedentop, Heinz

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N2 - 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.

AB - 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.

KW - Convexity

KW - concavity

KW - eigenvalues

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JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 5-6

ER -

Convexity and concavity of eigenvalue sums

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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