Karcher mean of three variables and quadric hypersurfaces

CHOI, HAYOUNG; LIM, YONGDO; GHIGLIONI, EDUARDO

Congreso; 2020 KMS spring meeting; 2020

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 talk we will present a recent study of the linear formula for the Karcher mean of 2×2 positive definite Hermitian matrices: Λ(A,B,C)=xA+yB+zC with \emph{nonnegative} coefficients, where the existence of nonnegative solutions is guaranteed by Sturm's SLLN and Holbrook's no dice theorem. On the other hand we will talk about the quadric surface induced by the determinantal formula: det(ABC)^(1/3)=det(xA+yB+zC). 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.