• TITLE: Riemannian geometry of Grassmann manifolds with a view on algorithmic computation
• AUTHORS: P.-A. Absil, R. Mahony, R. Sepulchre.
• ABSTRACT:
  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 $\rr^n$. 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.
• STATUS: Published in Acta Appl. Math., Volume 80, Issue 2, pp. 199-220, January 2004.
• DATE OF ENTRY: November 2002.

• Published version: doi:10.1023/B:ACAP.0000013855.14971.91.
• Preprint: PDF file.

BibTeX citation:

@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)},
}