Analyticity of the Karcher mean coefficient maps

Abstract

The main concern of this paper is the three-variable Karcher (alternatively, Riemannian or Cartan) matrix mean for 2×2 linearly independent triples of positive definite matrices. We show that such triples form an analytic manifold and obtain three analytic functions from (1,∞)2×(0,π/2) to (0,1) realizing the Karcher mean as a linear combination of variables.