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.
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Absil, PA., Mahony, R. & Sepulchre, R. Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation. Acta Applicandae Mathematicae 80, 199–220 (2004). https://doi.org/10.1023/B:ACAP.0000013855.14971.91
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DOI: https://doi.org/10.1023/B:ACAP.0000013855.14971.91