Abstract
A substantial theory of the Karcher mean exists in the settings of Riemannian manifolds and positive matrix and operator spaces. Here a general setting for the study of the Karcher mean on Lie groups is proposed. Local existence and uniqueness results already exist, but here, a significant global result is obtained. It is shown that a natural computable iteration scheme for the Karcher mean exists for the Lie group of upper triangular unipotent matrices and that it always converges starting from any point after finitely many steps.
Similar content being viewed by others
Data Availability
No data is available.
References
Ando, T.: Concavity of certain maps on positive definite matrices and applications to Hadamard products. Linear Algebra Appl. 26, 203–241 (1979)
Ando, T., Li, C.K., Mathias, R.: Geometric means. Linear Algebra Appl. 385, 305–334 (2004)
Bhatia, R., Holbrook, J.: Noncommutative geometric means. Math. Intell. 28, 32–39 (2006)
Choi, H., Kim, S., Lim, Y.: A binomial expansion formula for weighted geometric means of unipotent matrices. Linear Multilinear Algebra (2022) 72:4 615–630
Duan, X., Sun, H., Peng, L.: Riemannian means on special Euclidean group and unipotent matrices group. Sci. World J. 2013:292787 (2013)
Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509–541 (1977)
Kim, S., Mer, V.N.: The Wasserstein mean of unipotent matrices. Linear Multilinear Algebra (2023) (published online)
Lawson, J., Lim, Y.: Monotonic properties of the least squares mean. Math. Ann. 351, 267–279 (2011)
Moakher, M.: A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 26, 735–747 (2005)
Pennec, X., Arsigny, V.: Exponential barycenters of the canonical Cartan connection and invariant means on Lie groups. In: Barbaresco, F., Mishra, A., Nielsen, F. (eds.) Matrix Information Geometry, pp. 123–168. Springer, Berlin (2012)
Pusz, W., Woronowicz, S.L.: Functional calculus for sesquilinear forms and the purification map. Rep. Math. Phys. 8, 159–170 (1975)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Tin-Yau Tam.
To Chi-Kwong Li on his 65th birthday.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Lawson, J. Weighted Karcher means on unipotent Lie groups. Adv. Oper. Theory 9, 27 (2024). https://doi.org/10.1007/s43036-024-00326-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43036-024-00326-9