An Approach to the Log-Euclidean Mean via the Karcher Mean on Symmetric Cones

In a general symmetric cone, we show that certain sequence of the Karcher means converges to the Log-Euclidean mean by using the fact that the Karcher mean is the limit of inductive means. One can see this as a generalization of the Lie-Trotter formula of positive definite matrices into a symmetric...

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Bibliographic Details
Published inTaiwanese journal of mathematics Vol. 20; no. 1; pp. 191 - 203
Main Authors Kim, Sejong, Ji, Un Cig, Kum, Sangho
Format Journal Article
LanguageEnglish
Published Mathematical Society of the Republic of China 01.02.2016
Subjects
Algebra
Cones
Eigenvalues
Geometric mean
Inner products
Least squares
Mathematical theorems
Mathematical vectors
Real numbers
Symmetry
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Summary:In a general symmetric cone, we show that certain sequence of the Karcher means converges to the Log-Euclidean mean by using the fact that the Karcher mean is the limit of inductive means. One can see this as a generalization of the Lie-Trotter formula of positive definite matrices into a symmetric cone setting via the least squares mean. 2010Mathematics Subject Classification. 47A64, 17C50, 15B48, 53C20. Key words and phrases. Lie-Trotter formula, Least squares mean, Symmetric cone, Hadamard space.
ISSN:1027-5487
2224-6851
DOI:10.11650/tjm.20.2016.5559