Geometric means of structured matrices


The geometric mean of positive definite matrices is usually identified with the Karcher mean, which possesses all properties—generalized from the scalar case—a geometric mean is expected to satisfy. Unfortunately, the Karcher mean is typically not structure preserving, and destroys, e.g., Toeplitz and band structures, which emerge in many applications. For this reason, the Karcher mean is not always recommended for modeling averages of structured matrices. In this article a new definition of a geometric mean for structured matrices is introduced, its properties are outlined, algorithms for its computation, and numerical experiments are provided. In the Toeplitz case an existing mean based on the Kähler metric is analyzed for comparison.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. 1.

    Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)

    Google Scholar 

  2. 2.

    Ando, T., Li, C., Mathias, R.: Geometric means. Linear Algebra Appl. 385, 305–334 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Barbaresco, F.: Information intrinsic geometric flows. In: Mohammad-Djafari, A. (ed.) Bayesian Inference and Maximum Entropy Methods in Science and Engineering. American Institute of Physics Conference Series, vol. 872, pp. 211–218 (2006)

    Google Scholar 

  4. 4.

    Barbaresco, F.: Interactions between symmetric cone and information geometries: Bruhat-tits and Siegel spaces models for high resolution autoregressive Doppler imagery. In: Nielsen, F. (ed.) Emerging Trends in Visual Computing. Lecture Notes in Computer Science, vol. 5416, pp. 124–163. Springer, Berlin (2009)

    Google Scholar 

  5. 5.

    Barbaresco, F.: Robust statistical radar processing in Fréchet metric space: OS-HDR-CFAR and OS-STAP processing in Siegel homogeneous bounded domains. In: Proceedings of IRS’11, International Radar Conference, Leipzig (2011)

    Google Scholar 

  6. 6.

    Barbaresco, F.: Information geometry of covariance matrix: Cartan–Siegel homogeneous bounded domains, Mostow/Berger fibration and Fréchet median. In: Bhatia, R., Nielsen, F. (eds.) Matrix Information Geometry, pp. 199–256. Springer, Berlin (2012)

    Google Scholar 

  7. 7.

    Batchelor, P.G., Moakher, M., Atkinson, D., Calamante, F., Connelly, A.: A rigorous framework for diffusion tensor calculus. Magn. Reson. Med. 53, 221–225 (2005)

    Article  Google Scholar 

  8. 8.

    Bhatia, R.: Positive Definite Matrices. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2007)

    Google Scholar 

  9. 9.

    Bhatia, R., Holbrook, J.: Riemannian geometry and matrix geometric means. Linear Algebra Appl. 413(2–3), 594–618 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Bhatia, R., Karandikar, R.L.: Monotonicity of the matrix geometric mean. Math. Ann. 353(4), 1453–1467 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Bini, D.A., Iannazzo, B.: Computing the Karcher mean of symmetric positive definite matrices. Linear Algebra Appl. 438(4), 1700–1710 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Bini, D.A., Iannazzo, B.: A note on computing matrix geometric means. Adv. Comput. Math. 35(2–4), 175–192 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Bini, D., Pan, V.: Polynomial and Matrix Computations. Birkhäuser, Basel (1994)

    Google Scholar 

  14. 14.

    Bini, D., Meini, B., Poloni, F.: An effective matrix geometric mean satisfying the Ando–Li–Mathias properties. Math. Comput. 79(269), 437–452 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Bonnabel, S., Sepulchre, R.: Geometric distance and mean for positive semi-definite matrices of fixed rank. SIAM J. Matrix Anal. Appl. 31(3), 1055–1070 (2009)

    Article  MathSciNet  Google Scholar 

  16. 16.

    Bridson, M., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Springer, Berlin (1999)

    Google Scholar 

  17. 17.

    Cartan, E.: Leçons sur la géométrie des espaces de Riemann. Usp. Mat. Nauk 3(3), 218–222 (1948)

    Google Scholar 

  18. 18.

    Chu, M.T., Golub, G.H.: Structured inverse eigenvalue problems. Acta Numer. 11, 1–71 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Eidelman, Y., Gohberg, I.C.: On a new class of structured matrices. Integral Equ. Oper. Theory 34, 293–324 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. 20.

    Fiori, S.: Learning the Fréchet Mean over the Manifold of Symmetric Positive-Definite Matrices. Springer, Media (2009)

    Google Scholar 

  21. 21.

    Gaubert, S., Qu, Z.: The contraction rate in Thompson metric of order-preserving flows on a cone—application to generalized Riccati equations (2012). arXiv:1206.0448v1 [math.MG]

  22. 22.

    Higham, N.J.: Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia (2008)

    Google Scholar 

  23. 23.

    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  24. 24.

    Iannazzo, B.: The geometric mean of two matrices from a computational viewpoint (2011). arXiv:1201.0101v1 [math.NA]

  25. 25.

    Jeuris, B., Vandebril, R., Vandereycken, B.: A survey and comparison of contemporary algorithms for computing the matrix geometric mean. Electron. Trans. Numer. Anal. 39, 379–402 (2012)

    MathSciNet  Google Scholar 

  26. 26.

    Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30(5), 509–541 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  27. 27.

    Kendall, W.: Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence. Proc. Lond. Math. Soc. 61(2), 371–406 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  28. 28.

    Lang, S.: Fundamentals of Differential Geometry. Graduate Texts in Mathematics. Springer, Berlin (1999)

    Google Scholar 

  29. 29.

    Lapuyade-Lahorgue, J., Barbaresco, F.: Radar detection using Siegel distance between autoregressive processes, application to HF and X-band radar. In: IEEE Radar Conference, 2008. RADAR’08, Rome (2008)

    Google Scholar 

  30. 30.

    Lawson, J., Lim, Y.: The geometric mean, matrices, metrics, and more. Am. Math. Mon. 108(9), 797–812 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  31. 31.

    Lawson, J., Lim, Y.: Monotonic properties of the least squares mean. Math. Ann. 351(2), 267–279 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. 32.

    Lim, Y.: Riemannian and Finsler structures of symmetric cones. Trends Math. 4(2), 111–118 (2001)

    Google Scholar 

  33. 33.

    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)

    Article  MATH  MathSciNet  Google Scholar 

  34. 34.

    Moakher, M.: On the averaging of symmetric positive-definite tensors. J. Elast. 82(3), 273–296 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  35. 35.

    Nakamura, K.: Geometric means of positive operators. Kyungpook Math. J. 49, 167–181 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  36. 36.

    Pálfia, M.: The Riemann barycenter computation and means of several matrices. Int. J. Comput. Math. Sci. 3(3), 128–133 (2009)

    MathSciNet  Google Scholar 

  37. 37.

    Qi, C.: Numerical optimization methods on Riemannian manifolds. Ph.D. thesis, Florida State University, College of arts and sciences (2011)

  38. 38.

    Rentmeester, Q., Absil, P.-A.: Algorithm comparison for Karcher mean computation of rotation matrices and diffusion tensors. In: 19th Conference on European Signal Processing, 2011, EUSIPCO (2011)

    Google Scholar 

  39. 39.

    Thompson, A.C.: On certain contraction mappings in a partially ordered vector space. Proc. Am. Math. Soc. 14(3), 438–443 (1963)

    MATH  Google Scholar 

  40. 40.

    Vandebril, R., Van Barel, M., Mastronardi, N.: Matrix Computations and Semiseparable Matrices. Linear Systems, vol. 1. Johns Hopkins University Press, Baltimore (2008)

    Google Scholar 

  41. 41.

    Yang, L.: Medians of probability measures in Riemannian manifolds and applications to radar target detection. Ph.D. thesis, Université de Poitiers (2011)

  42. 42.

    Yang, L., Arnaudon, M., Barbaresco, F.: Geometry of covariance matrices and computation of median. In: AIP Conference Proceedings, 30th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, vol. 1305, pp. 479–486 (2011)

    Google Scholar 

Download references


The authors would like to thank the referees for many insightful comments which improved the presentations of the paper. In particular, they are indebted with a referee which provided the elegant proof for Theorem 2.1.

Author information



Corresponding author

Correspondence to Raf Vandebril.

Additional information

This work was partially supported by MIUR grant number 2002014121; by the Research Council KU Leuven, projects OT/11/055 (Spectral Properties of Perturbed Normal Matrices and their Applications), CoE EF/05/006 Optimization in Engineering (OPTEC); by the Fund for Scientific Research–Flanders (Belgium) project G034212N (Reestablishing Smoothness for Matrix Manifold Optimization via Resolution of Singularities); and by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office, Belgian Network DYSCO (Dynamical Systems, Control, and Optimization).

Communicated by Peter Benner.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bini, D.A., Iannazzo, B., Jeuris, B. et al. Geometric means of structured matrices. Bit Numer Math 54, 55–83 (2014).

Download citation


  • Matrix geometric mean
  • Structured matrices
  • Manifold optimization

Mathematics Subject Classification (2010)

  • 14Q10
  • 15A24
  • 15B05
  • 53B21
  • 65K10