TY - JOUR

T1 - The Karcher mean of three variables and quadric surfaces

AU - Choi, Hayoung

AU - Ghiglioni, Eduardo

AU - Lim, Yongdo

N1 - Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2020/10/15

Y1 - 2020/10/15

N2 - The Riemannian or Karcher mean has recently become an important tool for the averaging and study of positive definite matrices. Finding an explicit formula for the Karcher mean is problematic even for 2×2 triples. In this paper we study (1) the linear formula for the Karcher mean of 2×2 positive definite Hermitian matrices: Λ(A,B,C)=xA+yB+zC with nonnegative coefficients, where the existence of nonnegative solutions is guaranteed by Sturm's SLLN and Holbrook's no dice theorem, and (2) the quadric surface induced by the determinantal formula: [Formula presented]. We show that the solution set forms a simplex of dimension less than equal 2 and settle the first problem for linearly dependent case. A classification of the quadric surfaces from the linear form of Karcher means is presented in terms of linear (in)dependence of A,B,C: hyperboloid of two sheets, hyperbolic cylinder, and parallel planes.

AB - The Riemannian or Karcher mean has recently become an important tool for the averaging and study of positive definite matrices. Finding an explicit formula for the Karcher mean is problematic even for 2×2 triples. In this paper we study (1) the linear formula for the Karcher mean of 2×2 positive definite Hermitian matrices: Λ(A,B,C)=xA+yB+zC with nonnegative coefficients, where the existence of nonnegative solutions is guaranteed by Sturm's SLLN and Holbrook's no dice theorem, and (2) the quadric surface induced by the determinantal formula: [Formula presented]. We show that the solution set forms a simplex of dimension less than equal 2 and settle the first problem for linearly dependent case. A classification of the quadric surfaces from the linear form of Karcher means is presented in terms of linear (in)dependence of A,B,C: hyperboloid of two sheets, hyperbolic cylinder, and parallel planes.

KW - Karcher symmetric matrix

KW - Nussbaum's minimal geodesic

KW - Positive definite matrix

KW - Quadric surface

KW - Riemannian and Karcher mean

KW - Riemannian trace metric

UR - http://www.scopus.com/inward/record.url?scp=85087410120&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2020.124321

DO - 10.1016/j.jmaa.2020.124321

M3 - Article

AN - SCOPUS:85087410120

SN - 0022-247X

VL - 490

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

M1 - 124321

ER -