Advances in Computational Mathematics

, Volume 35, Issue 2–4, pp 175–192 | Cite as

A note on computing matrix geometric means

  • Dario Andrea Bini
  • Bruno Iannazzo


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!2 k ). 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.


Matrix geometric mean Matrix function Riemannian centroid Geodesic 

Mathematics Subject Classifications

65F30 15A15 


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  1. 1.
    Ando, T., Li, C.-K., Mathias, R.: Geometric means. Linear Algebra Appl. 385, 305–334 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bhatia, R.: Positive definite matrices. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2007)Google Scholar
  3. 3.
    Bhatia, R., Holbrook, J.: Riemannian geometry and matrix geometric means. Linear Algebra Appl. 413(2–3), 594–618 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bini, D., Iannazzo, B.: On the numerical solution of the matrix equation \(\sum_{i=1}^k \log({XA}_i^{-1})=\) 0. In: 16-th ILAS Conference, Pisa, 21–25 June 2010.
  5. 5.
    Bini, D.A., Iannazzo, B.: The Matrix Means Toolbox. Retrieved 7 May 2010.
  6. 6.
    Bini, D.A., Meini, B, Poloni, F.: An effective matrix geometric mean satisfying the Ando–Li–Mathias properties. Math. Comput. 79(269), 437–452 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Higham, N.J.: Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia (2008)Google Scholar
  8. 8.
    Lawson, J.D., Lim, Y.: The geometric mean, matrices, metrics, and more. Am. Math. Mon. 108(9), 797–812 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Manton, J.H.: A globally convergent numerical algorithm for computing the centre of mass on compact lie groups. In: Eighth International Conference on Control, Automation, Robotics and Vision, 2004. ICARCV 2004 8th, Kunming, China (2004)Google Scholar
  10. 10.
    Moakher, M.: A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 26(3), 735–747 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Moakher, M.: On the averaging of symmetric positive-definite tensors. J. Elast. 82(3), 273–296 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Poloni, F.: Constructing Matrix Geometric Means. arXiv:0906.3132v1 (2009)
  13. 13.
    Pusz, G., Woronowicz, S.L.: Functional calculus for sesquilinear forms and the purification map. Rep. Math. Phys. 8, 159–170 (1975)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.Dipartimento di Matematica e InformaticaUniversità di PerugiaPerugiaItaly

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