The metric d(A, B) = tr A + tr B − 2tr(A 1/2 BA 1/2 ) 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.

If A is a 2n × 2n real positive definite matrix, then there ex-) is a diagonal matrix with positive diagonal entries, which are called the symplectic eigenvalues of A. In this paper we derive several fundamental inequalities about these numbers. Among them are relations between the symplectic eigenvalues of A and those of A t , between the symplectic eigenvalues of m matrices A 1 , . . . , A m and of their Riemannian mean, a perturbation theorem, some variational principles, and some inequalities between the symplectic and ordinary eigenvalues.Sp (2n) = M ∈ M(2n) : M T JM = J .

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On the space of positive definite matrices we consider distance functions of the form d(A, B) = [trA(A, B) − trG(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) = A 1/2 B 1/2 this distance is A 1/2 − B 1/2 2 , and when G(A, B) = (A 1/2 BA 1/2 ) 1/2 it is the Bures-Wasserstein metric. We study two other cases: G(A, B) = A 1/2 (A −1/2 BA −1/2 ) 1/2 A 1/2 , the Pusz-Woronowicz geometric mean, and G(A, B) = exp log A+log B 2 , the log Euclidean mean. With these choices d(A, B) is no longer a metric, but it turns out that d 2 (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. √ 2 d(p, q) as the definition of the Hellinger distance. We have thenwhere A(p, q) is the arithmetic mean of the vectors p and q, G(p, q) is their geometric mean, and tr x stands for x i .A matrix/noncommutative/quantum version would seek to replace the probability vectors p and q by density matrices A and B ; i.e., positive semidefinite matrices A, B with tr A = tr B = 1. In the discussion that follows, the restriction on trace is not needed, and so we let A and B be any two positive semidefinite matrices. On the other hand, a part of our analysis 2010 Mathematics Subject Classification. 15B48, 49K35, 94A17, 81P45.

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