The metric d(A,B)=trA+trB−2tr(A1∕2BA1∕2)1∕2 1∕2 on the manifold of n×n positive definite matrices arises in various optimisation problems, in quantum information and in the theory of optimal transport. It is also related to Riemannian geometry. In the first part of this paper we study this metric from the perspective of matrix analysis, simplifying and unifying various proofs. Then we develop a theory of a mean of two, and a barycentre of several, positive definite matrices with respect to this metric. We explain some recent work on a fixed point iteration for computing this Wasserstein barycentre. Our emphasis is on ideas natural to matrix analysis. © 2018 Elsevier GmbH

Rajendra Bhatia

Department of Mathematics

T. Jain

Y. Lim

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Expositiones Mathematicae

On the space of positive definite matrices, we consider distance functions of the form d(A, B) = [tr A(A, B) - tr G(A, B)] 1 / 2, where A(A, B) is the arithmetic mean and G(A, B) is one of the different versions of the geometric mean. When G(A, B) = A1 / 2B1 / 2 this distance is ‖ A1 / 2- B1 / 2‖ 2, and when G(A,B)=(A1/2BA1/2)1/2 it is the Bures–Wasserstein metric. We study two other cases: G(A,B)=A1/2(A-1/2BA-1/2)1/2A1/2, the Pusz–Woronowicz geometric mean, and G(A,B)=exp(logA+logB2), the log Euclidean mean. With these choices, d(A, B) is no longer a metric, but it turns out that d2(A, B) is a divergence. We establish some (strict) convexity properties of these divergences. We obtain characterisations of barycentres of m positive definite matrices with respect to these distance measures. One of these leads to a new interpretation of a power mean introduced by Lim and Palfia, as a barycentre. The other uncovers interesting relations between the log Euclidean mean and relative entropy. © 2019, Springer Nature B.V.

Letters in Mathematical Physics

Matrix versions of the Hellinger distance

We consider the manifold of positive definite matrices endowed with the Fisher Riemannian metric and some other distances commonly used in information theory. We show that for each of them the best approximant to A from the unitary orbit of another matrix B commutes with A. © 2018 Elsevier Inc.

Linear Algebra and Its Applications

Procrustes problems in Riemannian manifolds of positive definite matrices

An attractive candidate for the geometric mean of m positive definite matrices A 1, ..., A m is their Riemannian barycentre G. One of its important operator theoretic properties, monotonicity in the m arguments, has been established recently by Lawson and Lim. We give an elementary proof of this property using standard matrix analysis and some counting arguments. We derive some new inequalities for G. One of these says that, for any unitarily invariant norm, {pipe}{pipe}{pipe}G{pipe}{pipe}{pipe} is not bigger than the geometric mean of {pipe}{pipe}{pipe}A 1{pipe}{pipe}{pipe}, ..., {pipe}{pipe}{pipe}A m{pipe}{pipe}{pipe}. © 2011 Springer-Verlag.

Mathematische Annalen

Monotonicity of the matrix geometric mean

The geometric mean of two positive definite matrices has been defined in several ways and studied by several authors, including Pusz and Woronowicz, and Ando. The characterizations by these authors do not readily extend to three matrices and it has been a long-standing problem to define a natural geometric mean of three positive definite matrices. In some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of A and B as the midpoint of the geodesic (with respect to a natural Riemannian metric) joining A and B. This suggests some natural definitions for a geometric mean of three positive definite matrices. We explain the necessary geometric background and explore the properties of some of these candidates. © 2004 Elsevier Inc. All rights reserved.

Journal	Data powered by TypesetExpositiones Mathematicae
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ISSN	07230869