Acta Applicandae Mathematica

, Volume 80, Issue 2, pp 199-220

First online:

Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation

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

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


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