The Karcher mean of three variables and quadric surfaces

The Riemannian or Karcher mean has recently become an important tool for the averaging and study of positive definite matrices. Finding an explicit formula for the Karcher mean is problematic even for 2×2 triples. In this paper we study (1) the linear formula for the Karcher mean of 2×2 positive definite Hermitian matrices: Λ(A,B,C)=xA+yB+zC with nonnegative coefficients, where the existence of nonnegative solutions is guaranteed by Sturm's SLLN and Holbrook's no dice theorem, and (2) the quadric surface induced by the determinantal formula: det⁡(ABC)13=det⁡(xA+yB+zC). We show that the solution set forms a simplex of dimension less than equal 2 and settle the first problem for linearly dependent case. A classification of the quadric surfaces from the linear form of Karcher means is presented in terms of linear (in)dependence of A,B,C: hyperboloid of two sheets, hyperbolic cylinder, and parallel planes.