A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices

Abstract. In this paper we introduce metric-based means for the space of positive-definite matrices. The mean associated with the Euclidean metric of the ambient space is the usual arithmetic mean. The mean associated with the Riemannian metric corresponds to the geometric mean. We discuss some invariance properties of the Riemannian mean and we use differential geometric tools to give a characterization of this mean. Key words. Geometric mean, Positive-definite symmetric matrices, Riemannian distance, Geodesics. AMS subject classifications. 47A64, 26E60, 15A48, 15A57