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- W2612291276 abstract "In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold $mathbb{G}_r(mathbb{R}^k)$ of linear subspaces of dimension $r<k$ in $mathbb{R}^k$ which avoids the use of equivalence classes. The set $mathbb{G}_r(mathbb{R}^k)$ is equipped with an atlas which provides it with the structure of an analytic manifold modelled on $mathbb{R}^{(k-r)times r}$. Then we define an atlas for the set $mathcal{M}_r(mathbb{R}^{k times r})$ of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $mathbb{G}_r(mathbb{R}^k)$ and typical fibre $mathrm{GL}_r$, the general linear group of invertible matrices in $mathbb{R}^{ktimes k}$. Finally, we define an atlas for the set $mathcal{M}_r(mathbb{R}^{n times m})$ of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $mathbb{G}_r(mathbb{R}^n) times mathbb{G}_r(mathbb{R}^m)$ and typical fibre $mathrm{GL}_r$. The atlas of $mathcal{M}_r(mathbb{R}^{n times m})$ is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set $mathcal{M}_r(mathbb{R}^{n times m})$ equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space $mathbb{R}^{n times m}$ equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space $mathbb{R}^{n times m}$, seen as the union of manifolds $mathcal{M}_r(mathbb{R}^{n times m})$, as an analytic manifold equipped with a topology for which the matrix rank is a continuous map." @default.
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- W2612291276 date "2017-05-12" @default.
- W2612291276 modified "2023-10-16" @default.
- W2612291276 title "Principal bundle structure of matrix manifolds" @default.
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