A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank
By Bart Vandereycken, Pierre-Antoine Absil, Stefan Vandewalle. 2013
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We present a homogeneous space geometry for the manifold of symmetric positive semidefinite matrices of fixed rank. The total space is the general linear group endowed with its natural rightinvariant metric, and the metric on the homogeneous space is chosen such that the quotient space is the image of a Riemannian submersion from the total space. As a result, we obtain complete geodesics that are the image of certain geodesics on the general linear group. We derive in addition an efficient closed-form expression for these geodesics. Furthermore, we isometrically embed the abstract quotient space into the space of real matrices. This allows us to interpret the vector fields, the metric and the geodesics in terms of concrete matrices. Finally, we compare our geometry with some other geometries in the literature. In particular, we show how other geodesics and their approximations relate to ours.
Subfields: Applied Mathematics, Computational Mechanics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/6da8439b1062d4156e6386196ccf7a22a29c2a8e
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In Applied Mathematics
In Computational Mechanics
In Computational Theory and Mathematics
Citations in the corpus
- Geometry of Logarithmic Strain Measures in Solid Mechanics (2015, methodology): “Using a similar proof, exactly the same result can be shown for the geodesic distance distgeod,right induced by the right-GL(n)-invariant, left-O(n)-invariant Riemannian metric [207]”