Title

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

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

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