The matrix geometric mean of parameterized, weighted arithmetic and harmonic means (Q551336)

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.