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.