Linearity of Cartan and Wasserstein means

초록

The convex cone of positive definite Hermitian matrices has two important Riemannian geometries, where the para-metrized weighted geometric (alternatively, Cartan) and Wasserstein means appear as the corresponding geodesics. The major problem with which this paper is concerned is lin-earity of Cartan and Wasserstein geodesics; when the Cartan (resp. Wasserstein) geodesic between two positive definite ma-trices A and B does lie in the space spanned by them and what the path in the plane to which it corresponds is. We settle the problem completely for the Cartan geometry and partially for the Wasserstein geometry. We show that their linearity problems for linearly independent A and B are equivalent to the solvability of the following equations for positive reals x and y with xy < 1/4 respectively (xA+ yB)(-1) = yA(-1) + xB(-1), (AB)(1/2 )+ (BA)(1/2)/ (2) = xA + y(B).

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