Document Type
Article
Publication Date
11-1-2011
Abstract
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 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 Lμ≤Lν 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.© 2011 Elsevier Inc. All rights reserved.
Publication Source (Journal or Book title)
Linear Algebra and Its Applications
First Page
2114
Last Page
2131
Recommended Citation
Kim, S., Lawson, J., & Lim, Y. (2011). The matrix geometric mean of parameterized, weighted arithmetic and harmonic means. Linear Algebra and Its Applications, 435 (9), 2114-2131. https://doi.org/10.1016/j.laa.2011.04.010