Acta Applicandae Mathematica

, Volume 80, Issue 2, pp 199–220 | Cite as

Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation

  • P.-A. Absil
  • R. Mahony
  • R. Sepulchre


We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R n . In these formulas, p-planes are represented as the column space of n×p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications – computing an invariant subspace of a matrix and the mean of subspaces – are worked out.

Grassmann manifold noncompact Stiefel manifold principal fiber bundle Levi-Civita connection parallel transportation geodesic Newton method invariant subspace mean of subspaces 


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  1. 1.
    Absil, P.-A.: Invariant subspace computation: A geometric approach, Ph.D. thesis, Faculté des Sciences Appliquées, Université de Liège, Secrétariat de la FSA, Chemin des Chevreuils 1 (Bât. B52), 4000 Liège, Belgium, 2003.Google Scholar
  2. 2.
    Absil, P.-A., Mahony, R., Sepulchre, R. and Van Dooren, P.: A Grassmann–Rayleigh quotient iteration for computing invariant subspaces, SIAM Rev. 44(1) (2002), 57–73.Google Scholar
  3. 3.
    Absil, P.-A., Sepulchre, R., Van Dooren, P. and Mahony, R.: Cubically convergent iterations for invariant subspace computation, SIAM J. Matrix Anal. Appl., to appear.Google Scholar
  4. 4.
    Absil, P.-A. and Van Dooren, P.: Two-sided Grassmann–Rayleigh quotient iteration, SIAM J. Matrix Anal. Appl. (2002), submitted.Google Scholar
  5. 5.
    Björk, Å. and Golub, G. H.: Numerical methods for computing angles between linear subspaces, Math. Comp. 27 (1973), 579–594.Google Scholar
  6. 6.
    Boothby, W. M.: An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, 1975.Google Scholar
  7. 7.
    Chatelin, F.: Simultaneous Newton's iteration for the eigenproblem, Computing 5(Suppl.) (1984), 67–74.Google Scholar
  8. 8.
    Chavel, I.: Riemannian Geometry – A Modern Introduction, Cambridge Univ. Press, 1993.Google Scholar
  9. 9.
    Common, P. and Golub, G. H.: Tracking a few extreme singular values and vectors in signal processing, Proc. IEEE 78(8) (1990), 1327–1343.Google Scholar
  10. 10.
    Demmel, J. W.: Three methods for refining estimates of invariant subspaces, Computing 38 (1987), 43–57.Google Scholar
  11. 11.
    Dennis, J. E. and Schnabel, R. B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall Series in Computational Mathematics, Prentice-Hall, Englewood Cliffs, NJ, 1983.Google Scholar
  12. 12.
    do Carmo, M. P.: Riemannian Geometry, Birkhäuser, 1992.Google Scholar
  13. 13.
    Doolin, B. F. and Martin, C. F.: Introduction to Differential Geometry for Engineers, Monographs and Textbooks in Pure and Applied Mathematics 136, Marcel Deckker, Inc., New York, 1990.Google Scholar
  14. 14.
    Edelman, A., Arias, T. A. and Smith, S. T.: The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl. 20(2) (1998), 303–353.Google Scholar
  15. 15.
    Ferrer, J., García, M. I. and Puerta, F.: Differentiable families of subspaces, Linear Algebra Appl. 199 (1994), 229–252.Google Scholar
  16. 16.
    Gabay, D.: Minimizing a differentiable function over a differential manifold, J. Optim. Theory Appl. 37(2) (1982), 177–219.Google Scholar
  17. 17.
    Golub, G. H. and Van Loan, C. F.: Matrix Computations, 3rd edn, The Johns Hopkins Univ. Press, 1996.Google Scholar
  18. 18.
    Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math. 80, Academic Press, Oxford, 1978.Google Scholar
  19. 19.
    Helmke, U. and Moore, J. B.: Optimization and Dynamical Systems, Springer, 1994.Google Scholar
  20. 20.
    Karcher, H.: Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30(5) (1977), 509–541.Google Scholar
  21. 21.
    Kendall, W. S.: Probability, convexity, and harmonic maps with small image. I. Uniqueness and fine existence, Proc. London Math. Soc. 61(2) (1990), 371–406.Google Scholar
  22. 22.
    Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Vol. 1, 2, Wiley, 1963.Google Scholar
  23. 23.
    Leichtweiss, K.: Zur riemannschen Geometrie in grassmannschen Mannigfaltigkeiten, Math. Z. 76 (1961), 334–366.Google Scholar
  24. 24.
    Luenberger, D. G.: Optimization by Vector Space Methods, Wiley, Inc., 1969.Google Scholar
  25. 25.
    Lundström, E. and Eldén, L.: Adaptive eigenvalue computations using Newton's method on the Grassmann manifold, SIAM J. Matrix Anal. Appl. 23(3) (2002), 819–839.Google Scholar
  26. 26.
    Machado, A. and Salavessa, I.: Grassmannian manifolds as subsets of Euclidean spaces, Res. Notes in Math. 131 (1985), 85–102.Google Scholar
  27. 27.
    Mahony, R. E.: The constrained Newton method on a Lie group and the symmetric eigenvalue problem, Linear Algebra Appl. 248 (1996), 67–89.Google Scholar
  28. 28.
    Mahony, R. and Manton, J. H.: The geometry of the Newton method on non-compact Lie groups, J. Global Optim. 23(3) (2002), 309–327.Google Scholar
  29. 29.
    Nomizu, K.: Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33–65.Google Scholar
  30. 30.
    Ran, A. C. M. and Rodman, L.: A class of robustness problems in matrix analysis, In: D. Alpay, I. Gohberg, and V. Vinnikov (eds), Interpolation Theory, Systems Theory and Related Topics, The Harry Dym Anniversary Volume, Operator Theory: Advances and Applications 134, Birkhäuser, 2002, pp. 337–383.Google Scholar
  31. 31.
    Simoncini, V. and Elden, L.: Inexact Rayleigh quotient-type methods for eigen-value computations, BIT 42(1) (2002), 159–182.Google Scholar
  32. 32.
    Smith, S. T.: Geometric optimization methods for adaptive filtering, Ph.D. thesis, Division of Applied Sciences, Harvard University, Cambridge, MA, 1993.Google Scholar
  33. 33.
    Smith, S. T.: Optimization techniques on Riemannian manifolds, In: A. Bloch (ed.), Hamiltonian and Gradient Flows, Algorithms and Control, Fields Institute Communications, Vol. 3, Amer. Math. Soc., 1994, pp. 113–136.Google Scholar
  34. 34.
    Stewart, G. W.: Error and perturbation bounds for subspaces associated with certain eigen-value problems, SIAM Rev. 15(4) (1973), 727–764.Google Scholar
  35. 35.
    Udri¸ste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds, Kluwer Acad. Publ., Dordrecht, 1994.Google Scholar
  36. 36.
    Wong, Y.-C.: Differential geometry of Grassmann manifolds, Proc. Nat. Acad. Sci. U.S.A. 57 (1967), 589–594.Google Scholar
  37. 37.
    Woods, R. P.: Characterizing volume and surface deformations in an atlas framework: Theory, applications, and implementation, NeuroImage 18(3) (2003), 769–788.Google Scholar

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© Kluwer Academic Publishers 2004

Authors and Affiliations

  • P.-A. Absil
  • R. Mahony
  • R. Sepulchre

There are no affiliations available

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