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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 $C♯D$ is the metric midpoint of the of arithmetic mean $A=\frac{1}{2}\left(C+D\right)$ and the harmonic mean $H=2{\left({C}^{-1}+{D}^{-1}\right)}^{-1}$ 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 $\omega \in {\left(0,1\right)}^{m}$ with ${\parallel \omega \parallel }_{1}=1$ and positive definite matrices ${A}_{1},\cdots ,{A}_{m}$ with $A=\left({A}_{1},\cdots ,{A}_{m}\right)$ they introduce the weighted $A♯H$-mean to be the matrix geometric mean of the weighted arithmetic and harmonic means: $ℒ\left(\omega ;A\right):=\left({\sum }_{i}{\omega }_{i}{A}_{i}\right)♯{\left({\sum }_{i}{\omega }_{i}{A}_{i}^{-1}\right)}^{-1}$. Many properties of this weighted mean are presented, and it is interpreted via the Kullback-Leibler divergence from probability theory and information theory.
##### MSC:
 15B48 Positive matrices and their generalizations; cones of matrices 47A64 Operator means, shorted operators, etc. 26E60 Means
##### References:
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