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The matrix geometric mean of parameterized, weighted arithmetic and harmonic means. (English) Zbl 1222.15036
For positive definite matrices $C$ and $D$, the matrix geometric mean $C \sharp D$ is the metric midpoint of the of arithmetic mean $A = \frac12(C + D)$ and the harmonic mean $H = 2(C^{-1} + D^{-1})^{-1}$ for the trace metric. The authors consider the more general construction of taking the geometric mean of the weighted $n$-variable arithmetic and harmonic means. More precisely, for $\omega \in (0, 1)^m$ with $\|\omega\|_1=1$ and positive definite matrices $A_1,\dots,A_m$ with $A = (A_1,\dots,A_m)$ they introduce the weighted $A\sharp H$-mean to be the matrix geometric mean of the weighted arithmetic and harmonic means: $\mathcal L(\omega;A) := \left( \sum_i \omega_i A_i\right) \sharp \left( \sum_i \omega_i A_i^{-1}\right)^{-1}$. Many properties of this weighted mean are presented, and it is interpreted via the Kullback-Leibler divergence from probability theory and information theory.

15B48Positive matrices and their generalizations; cones of matrices
47A64Operator means, shorted operators, etc.
Full Text: DOI
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