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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.

MSC:
 15A45 Miscellaneous inequalities involving matrices 15A48 Positive matrices and their generalizations (MSC2000) 53B21 Methods of Riemannian geometry 53C22 Geodesics 26E60 Means