Acta Applicandae Mathematica

, Volume 80, Issue 2, pp 199–220

Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation

Authors

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

DOI: 10.1023/B:ACAP.0000013855.14971.91

Cite this article as:
Absil, P., Mahony, R. & Sepulchre, R. Acta Applicandae Mathematicae (2004) 80: 199. doi:10.1023/B:ACAP.0000013855.14971.91

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 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

Copyright information

© Kluwer Academic Publishers 2004