Principal Bundle Structure of Matrix Manifolds
| Autor: | Anthony Nouy, Antonio Falcó, Marie Billaud-Friess |
|---|---|
| Přispěvatelé: | Institut de Recherche en Génie Civil et Mécanique (GeM), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS), Departamento de Ciencias, Físicas, Matemáticas y de la Computación, Universidad CEU Cardenal Herrera, Producción Científica UCH 2021, UCH. Departamento de Matemáticas, Física y Ciencias Tecnológicas |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Grassmann
Variedades de Mathematics - Differential Geometry Pure mathematics Differential topology matrix manifolds Rank (linear algebra) General Mathematics 010103 numerical & computational mathematics 02 engineering and technology 01 natural sciences Grassmann manifolds Matrix (mathematics) Manifolds (Mathematics) Variedades (Matemáticas) 020204 information systems Grassmannian FOS: Mathematics 0202 electrical engineering electronic engineering information engineering Computer Science (miscellaneous) QA1-939 Geometría diferencial Mathematics - Numerical Analysis 0101 mathematics Mathematics::Symplectic Geometry Engineering (miscellaneous) Mathematics Grassmann manifold principal bundles Atlas (topology) Numerical Analysis (math.NA) Geometry Differential low-rank matrices Submanifold Principal bundle Manifold Analytic manifold Differential Geometry (math.DG) [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] Mathematics::Differential Geometry 15A03 15A23 55R10 65F99 Topología diferencial [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] |
| Zdroj: | Mathematics, Vol 9, Iss 1669, p 1669 (2021) CEU Repositorio Institucional Fundación Universitaria San Pablo CEU (FUSPCEU) Mathematics Volume 9 Issue 14 |
| ISSN: | 2227-7390 |
| Popis: | 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 Gr(Rk) of linear subspaces of dimension r< k in Rk, which avoids the use of equivalence classes. The set Gr(Rk) is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on R(k−r)×r. Then, we define an atlas for the set Mr(Rk×r) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rk) and typical fibre GLr, the general linear group of invertible matrices in Rk×k. Finally, we define an atlas for the set Mr(Rn×m) of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rn)×Gr(Rm) and typical fibre GLr. The atlas of Mr(Rn×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 Mr(Rn×m) equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space Rn×m equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space Rn×m, seen as the union of manifolds Mr(Rn×m), as an analytic manifold equipped with a topology for which the matrix rank is a continuous map. |
| Databáze: | OpenAIRE |
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