Article

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

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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