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The geometric mean, matrices, metrics, and more. (English) Zbl 1040.15016
Starting from eight characterizations of the geometric mean $$x = \sqrt{ab}$$ of two positive real numbers $$a$$ and $$b$$, this survey paper gives an overview of six extensions of this concept to the geometric mean of two positive definite matrices, to matrix orders, to Bruhat-Tits metric spaces, to differential matrix Riccati equations, to convex programming, and to the geometric mean of symmetric cones in Jordan algebras. All is done in light of the original eight definitions of the mean.

##### MSC:
 15A45 Miscellaneous inequalities involving matrices 26D07 Inequalities involving other types of functions 26E60 Means 15B48 Positive matrices and their generalizations; cones of matrices 17C37 Associated geometries of Jordan algebras 20G20 Linear algebraic groups over the reals, the complexes, the quaternions 46B20 Geometry and structure of normed linear spaces 90C25 Convex programming
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