Abstract
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(n 3 k 2) arithmetic operations per step whereas the methods based on the symmetrization technique of Ando et al. (Linear Algebra Appl 385:305–334, 2004) have complexity O(n 3 k!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 ten properties stated by Ando, Li and Mathias; a counterexample shows that monotonicity is not fulfilled.
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Communicated by Juan Manuel Peña and Rafael Bru.
Supported by PRIN 20083KLJEZ.
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Bini, D.A., Iannazzo, B. A note on computing matrix geometric means. Adv Comput Math 35, 175–192 (2011). https://doi.org/10.1007/s10444-010-9165-0
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DOI: https://doi.org/10.1007/s10444-010-9165-0