zbMATH — the first resource for mathematics

Riemannian geometry and matrix geometric means. (English) Zbl 1088.15022
The goals of this article are partly expository but include an analysis of recent approaches to the definition of a geometric mean for three (or more) positive definite matrices. The authors first review some standard constructions from Riemannian geometry. Then they show how these constructions lead to a better and deep understanding of the geometric mean of two positive definite matrices. This allows them to treat the problem of extending these ideas to three matrices.
The authors notice that the several definitions of geometric means appeared in the literature do not readily extend to three matrices. In some recent papers the geometric mean of \(A\) and \(B\) is explained as the midpoint of the geodesic (with respect to a natural Riemannian metric) joining \(A\) and \(B\). This new understanding of the geometric mean suggests some natural definitions for a geometric mean of three positive definite matrices.

15A45 Miscellaneous inequalities involving matrices
15B48 Positive matrices and their generalizations; cones of matrices
53B21 Methods of local Riemannian geometry
53C22 Geodesics in global differential geometry
26E60 Means
Full Text: DOI
[1] Ando, T., Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear algebra appl., 26, 203-241, (1979) · Zbl 0495.15018
[2] Ando, T.; Li, C.-K.; Mathias, R., Geometric means, Linear algebra appl., 385, 305-334, (2004) · Zbl 1063.47013
[3] Berger, M., A panoramic view of Riemannian geometry, (2003), Springer · Zbl 1038.53002
[4] Bhatia, R., Matrix analysis, (1997), Springer
[5] Bhatia, R., On the exponential metric increasing property, Linear algebra appl., 375, 211-220, (2003) · Zbl 1052.15013
[6] Bridson, M.; Haefliger, A., Metric spaces of non-positive curvature, (1999), Springer · Zbl 0988.53001
[7] Corach, G.; Porta, H.; Recht, L., Geodesics and operator means in the space of positive operators, Int. J. math., 4, 193-202, (1993) · Zbl 0809.47017
[8] Lawson, J.D.; Lim, Y., The geometric Mean, matrices, metrics, and more, Amer. math. monthly, 108, 797-812, (2001) · Zbl 1040.15016
[9] Moakher, M., A differential geometric approach to the geometric Mean of symmetric positive-definite matrices, SIAM J. matrix anal. appl., 26, 735-747, (2005) · Zbl 1079.47021
[10] M. Moakher, On the averaging of symmetric positive-definite tensors, 2004, preprint. · Zbl 1094.74010
[11] D. Petz, R. Temesi, Means of positive numbers and matrices, 2005, preprint. · Zbl 1108.47020
[12] Pusz, W.; Woronowicz, S.L., Functional calculus for sesquilinear forms and the purification map, Reports math. phys., 8, 159-170, (1975) · Zbl 0327.46032
[13] Trapp, G.E., Hermitian semidefinite matrix means and related matrix inequalities—an introduction, Linear and multilinear algebra, 16, 113-123, (1984) · Zbl 0548.15013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.