Matrix versions of the Hellinger distance

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 parallel to A(1/2)-B(1/2 parallel to)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.