Hyperbolicity of the Karcher mean

Eduardo Ghiglioni; Yongdo Lim

doi:10.1016/j.laa.2022.02.018

Hyperbolicity of the Karcher mean

Eduardo Ghiglioni
, Yongdo Lim

Department of Mathematics

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

2 Scopus citations

Abstract

The main concern of this paper is the Karcher mean of linearly independent triples (A,B,C) on the hyperbolic manifold of 2×2 positive definite matrices of determinant 1. We show that the Karcher mean is of the form Λ(A,B,C)=xA+y(B+C),0<x,yandx+2y<1 under the trace condition tr(AB⁻¹)=tr(AC⁻¹). We further find an invertible hyperbolic matrix M depending only on the trace values tr(AB⁻¹) and tr(BC⁻¹) such that [xy]=M[cosh⁡θsinh⁡θ] for some (unique) θ∈R.

Original language	English
Pages (from-to)	196-217
Number of pages	22
Journal	Linear Algebra and Its Applications
Volume	643
DOIs	https://doi.org/10.1016/j.laa.2022.02.018
State	Published - 15 Jun 2022

Keywords

CS-decomposition
Hyperbolic matrix
Hyperboloid of two sheets
Karcher mean
Positive definite matrix
Positively stable matrix

Access to Document

10.1016/j.laa.2022.02.018

Cite this

@article{d2c5b40a834a4bce92d23e796ba5b8f5,

title = "Hyperbolicity of the Karcher mean",

abstract = "The main concern of this paper is the Karcher mean of linearly independent triples (A,B,C) on the hyperbolic manifold of 2×2 positive definite matrices of determinant 1. We show that the Karcher mean is of the form Λ(A,B,C)=xA+y(B+C),0−1)=tr(AC−1). We further find an invertible hyperbolic matrix M depending only on the trace values tr(AB−1) and tr(BC−1) such that [xy]=M[cosh⁡θsinh⁡θ] for some (unique) θ∈R.",

keywords = "CS-decomposition, Hyperbolic matrix, Hyperboloid of two sheets, Karcher mean, Positive definite matrix, Positively stable matrix",

author = "Eduardo Ghiglioni and Yongdo Lim",

note = "Publisher Copyright: {\textcopyright} 2022 Elsevier Inc.",

year = "2022",

month = jun,

day = "15",

doi = "10.1016/j.laa.2022.02.018",

language = "English",

volume = "643",

pages = "196--217",

journal = "Linear Algebra and Its Applications",

issn = "0024-3795",

publisher = "Elsevier Inc.",

}

TY - JOUR

T1 - Hyperbolicity of the Karcher mean

AU - Ghiglioni, Eduardo

AU - Lim, Yongdo

PY - 2022/6/15

Y1 - 2022/6/15

N2 - The main concern of this paper is the Karcher mean of linearly independent triples (A,B,C) on the hyperbolic manifold of 2×2 positive definite matrices of determinant 1. We show that the Karcher mean is of the form Λ(A,B,C)=xA+y(B+C),0−1)=tr(AC−1). We further find an invertible hyperbolic matrix M depending only on the trace values tr(AB−1) and tr(BC−1) such that [xy]=M[cosh⁡θsinh⁡θ] for some (unique) θ∈R.

AB - The main concern of this paper is the Karcher mean of linearly independent triples (A,B,C) on the hyperbolic manifold of 2×2 positive definite matrices of determinant 1. We show that the Karcher mean is of the form Λ(A,B,C)=xA+y(B+C),0−1)=tr(AC−1). We further find an invertible hyperbolic matrix M depending only on the trace values tr(AB−1) and tr(BC−1) such that [xy]=M[cosh⁡θsinh⁡θ] for some (unique) θ∈R.

KW - CS-decomposition

KW - Hyperbolic matrix

KW - Hyperboloid of two sheets

KW - Karcher mean

KW - Positive definite matrix

KW - Positively stable matrix

UR - https://www.scopus.com/pages/publications/85126783666

U2 - 10.1016/j.laa.2022.02.018

DO - 10.1016/j.laa.2022.02.018

M3 - Article

AN - SCOPUS:85126783666

SN - 0024-3795

VL - 643

SP - 196

EP - 217

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Hyperbolicity of the Karcher mean

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this