Log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter

Fumio Hiai; Yongdo Lim

doi:10.1016/j.laa.2018.04.029

Log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter

Fumio Hiai
, Yongdo Lim

Department of Mathematics

Tohoku University

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

3 Scopus citations

Abstract

In this paper we show that the eigenvalue map and the symplectic eigenvalue map of positive definite matrices are Lipschitz for the Cartan–Hadamard Riemannian metric, and establish log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter of integrable probability Borel measures. This leads a version of Jensen's inequality for geometric integrals of matrix-valued integrable random variables.

Original language	English
Pages (from-to)	129-144
Number of pages	16
Journal	Linear Algebra and Its Applications
Volume	553
DOIs	https://doi.org/10.1016/j.laa.2018.04.029
State	Published - 15 Sep 2018

Keywords

(Symplectic) eigenvalue
Cartan barycenter
Log-majorization
Positive definite matrix
Probability measure
Riemannian trace metric

Access to Document

10.1016/j.laa.2018.04.029

Cite this

@article{a24d540c0dc0479b83185e0e44a13be1,

title = "Log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter",

abstract = "In this paper we show that the eigenvalue map and the symplectic eigenvalue map of positive definite matrices are Lipschitz for the Cartan–Hadamard Riemannian metric, and establish log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter of integrable probability Borel measures. This leads a version of Jensen's inequality for geometric integrals of matrix-valued integrable random variables.",

keywords = "(Symplectic) eigenvalue, Cartan barycenter, Log-majorization, Positive definite matrix, Probability measure, Riemannian trace metric",

author = "Fumio Hiai and Yongdo Lim",

note = "Publisher Copyright: {\textcopyright} 2018 Elsevier Inc.",

year = "2018",

month = sep,

day = "15",

doi = "10.1016/j.laa.2018.04.029",

language = "English",

volume = "553",

pages = "129--144",

journal = "Linear Algebra and Its Applications",

issn = "0024-3795",

publisher = "Elsevier Inc.",

}

TY - JOUR

T1 - Log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter

AU - Hiai, Fumio

AU - Lim, Yongdo

PY - 2018/9/15

Y1 - 2018/9/15

N2 - In this paper we show that the eigenvalue map and the symplectic eigenvalue map of positive definite matrices are Lipschitz for the Cartan–Hadamard Riemannian metric, and establish log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter of integrable probability Borel measures. This leads a version of Jensen's inequality for geometric integrals of matrix-valued integrable random variables.

AB - In this paper we show that the eigenvalue map and the symplectic eigenvalue map of positive definite matrices are Lipschitz for the Cartan–Hadamard Riemannian metric, and establish log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter of integrable probability Borel measures. This leads a version of Jensen's inequality for geometric integrals of matrix-valued integrable random variables.

KW - (Symplectic) eigenvalue

KW - Cartan barycenter

KW - Log-majorization

KW - Positive definite matrix

KW - Probability measure

KW - Riemannian trace metric

UR - https://www.scopus.com/pages/publications/85046811346

U2 - 10.1016/j.laa.2018.04.029

DO - 10.1016/j.laa.2018.04.029

M3 - Article

AN - SCOPUS:85046811346

SN - 0024-3795

VL - 553

SP - 129

EP - 144

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this