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.

Rajendra Bhatia

Department of Mathematics

M. Congedo

Fulltext

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Linear Algebra and Its Applications

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

Expositiones Mathematicae

On the Bures–Wasserstein distance between positive definite matrices

Proceedings of the American Mathematical Society

Positive definite matrices and the S-divergence

Physical Review A

Quantum state discrimination: A geometric approach

Procrustes problems in Riemannian manifolds of positive definite matrices

Journal	Data powered by TypesetLinear Algebra and Its Applications
Publisher	Data powered by TypesetElsevier Inc.
ISSN	00243795