Paper ID sheet UCL-INMA-2004.901


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 of $\mathbb{R}^n$. In these formulas, $p$-planes are represented as the column space of $n\times p$ matrices. The Newton method on abstract Riemannian manifolds proposed by S. T. 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.
Key words
Published in Acta Appl. Math., Volume 80, Issue 2, pp. 199-220, January 2004
BibTeX entry

@article {AMS2004-01,
    AUTHOR = {Absil, P.-A. and Mahony, R. and Sepulchre, R.},
   FAUTHOR = {Absil, P.-A. and Mahony, R. and Sepulchre, R.},
     TITLE = {Riemannian geometry of {G}rassmann manifolds with a view on
              algorithmic computation},
   JOURNAL = {Acta Appl. Math.},
  FJOURNAL = {Acta Applicandae Mathematicae. An International Survey Journal
              on Applying Mathematics and Mathematical Applications},
    VOLUME = {80},
      YEAR = {2004},
    NUMBER = {2},
     PAGES = {199--220},
      ISSN = {0167-8019},
     CODEN = {AAMADV},
   MRCLASS = {53C30 (53C05 65J05)},
While the equation that follows the proof of Theorem 4.2 is correct, this is not the expression that one immediately gets when one writes the Newton equation (23) for a gradient vector field. In view of (14), there should be a Y^TY factor on the right-hand side as well as in the derivative on the left-hand side. However, observe that the derivative of function Y\mapsto Y^TY along the lift of \eta is zero since the lift of \eta belongs to the horizontal space H_Y defined in (8). Thus finally the Y^TY factor appears as a right multiplication on the left-hand side, and since it is invertible, it cancels with the same factor of the right-hand side. The equation is thus correct as written. It can be further simplified by noting that the projection inside the derivative on the left-hand side and the projection on the right-hand side are redundant since the gradient of the lifted function is already horizontal. Thanks to Daniel Karrasch for pointing this out.