Paper ID sheet - Systems and Control Engineering at ULg
• 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.

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},
MRCLASS = {53C30 (53C05 65J05)},
}


Errata:

While the equation that follows the proof of Theorem 4.2 is correct, this is not the expression that one immediately gets when one writes the Newton equation (23) for a gradient vector field. In view of (14), there should be a Y^TY factor on the right-hand side as well as in the derivative on the left-hand side. However, observe that the derivative of function Y\mapsto Y^TY along the lift of \eta is zero since the lift of \eta belongs to the horizontal space H_Y defined in (8). Thus finally the Y^TY factor appears as a right multiplication on the left-hand side, and since it is invertible, it cancels with the same factor of the right-hand side. The equation is thus correct as written. It can be further simplified by noting that the projection inside the derivative on the left-hand side and the projection on the right-hand side are redundant since the gradient of the lifted function is already horizontal. Thanks to Daniel Karrasch for pointing this out.
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