Go back to

• Abstracts of the Talks
• Program

Fumio Hiai  (Tohoku University, Japan)
Wednesday, August 04, 2010, room Hörsaal 1
Riemannian metrics on positive definite matrices related to means

This is a joint work with D. Petz and a continuation of the paper in LAA 430 (2009) with the same title. The Hermitian matrices form the -dimensional Euclidean space with respect to Hilbert-Schmidt inner product. The set of all positive definite matrices, being an open subset of , is naturally equipped with a manifold structure. An smooth kernel function induces a Riemannian metric on defined by

where is the spectral decomposition. Certain mportant quantities in quantum information geometry, such as statistical metric, quantum Fisher informations, and quantum variances, are Riemannian metrics arising from kernel functions of the form , a ()-power of a symmetric homogenous mean M(x,y) of x,y>0. We discuss the following topics concerning geodesic shortest curves and geodesic distance of Riemannian metrics on of this type.

(1) Since the Riemannian manifold with is complete if and only if , the existence of geodesic shortest curves in the case does not seem obvious. When A,B are commuting, we present an explicit formula of a geodesic shortest curve between A,B that is depending on but independent of the choice of M. Moreover, we show the existence of a geodesic shortest curve joining for the metric with if is sufficiently near 2.

(2) We present a necessary and sufficient condition for Riemannian metrics and induced by and to be isometric under the transformation given by a smooth function . The condition is explicitly given in terms of , and .

(3) From the above (2) we can construct a one-parameter isometric family of Riemannian metrics starting from any inside the set of Riemannian metrics we are treating. Those isometric families have different features between the cases and . We see that each of those families converges to the metric induced by the square of the logarithmic mean . Thus may be regarded as a (unique) attractor in the set of all Riemannian metrics of our discussion. From the fact that the geodesic shortest curve for the metric is (), this shows the Riemannian geometric interpretation for the limit formulas such as

where is the t-power mean ().