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

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