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

We define a new family of matrix means {L_μ(ω;A)}_μ∈R where ω and A vary over all positive probability vectors in R^m and m-tuples of positive definite matrices resp. Each of these means interpolates between the weighted harmonic mean (μ=-∞) and the arithmetic mean of the same weight (μ=∞) with L_μ≤L_ν for μ≤ν. Each has a variational characterization as the unique minimizer of the weighted sum for the symmetrized, parameterized Kullback-Leibler divergence. Furthermore, each can be realized as the common limit of the mean iteration by arithmetic and harmonic means (in the unparameterized case), or, more generally, the arithmetic and resolvent means. Other basic typical properties for a multivariable mean are derived.