A note on computing matrix geometric means

A new definition is introduced for the matrix geometric mean of a set of k positive definite n×n matrices together with an iterative method for its computation. The iterative method is locally convergent with cubic convergence and requires O(n3k2) arithmetic operations per step whereas the methods based on the symmetrization technique of Ando, Li and Mathias [Linear Algebra Appl., 385 (2004), pp. 305–334] have complexity O(n3k!2k). The new mean is obtained from the properties of the centroid of a triangle rephrased in terms of geodesics in a suitable Riemannian geometry on the set of positive definite matrices. It satisfies most part of the 10 properties stated by Ando, Li and Mathias; a counterexample shows that monotonicity is not fulfilled.