Principal Bundle Structure of Matrix Manifolds

Marie Billaud-Friess, Antonio Falcó and Anthony Nouy
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Marie Billaud-Friess: Department of Computer Science and Mathematics, Ecole Centrale de Nantes, 1 Rue de la Noë, BP 92101, CEDEX 3, 44321 Nantes, France
Antonio Falcó: Departamento de Matemáticas, Física y Ciencias Tecnológicas, Universidad CEU Cardenal Herrera, CEU Universities, San Bartolomé 55, 46115 Alfara del Patriarca, Spain
Anthony Nouy: Department of Computer Science and Mathematics, Ecole Centrale de Nantes, 1 Rue de la Noë, BP 92101, CEDEX 3, 44321 Nantes, France

Mathematics, 2021, vol. 9, issue 14, 1-17

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 G r ( R k ) of linear subspaces of dimension r < k in R k , which avoids the use of equivalence classes. The set G r ( R k ) 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 M r ( R k × r ) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base G r ( R k ) and typical fibre GL r , the general linear group of invertible matrices in R k × k . Finally, we define an atlas for the set M r ( R n × m ) of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base G r ( R n ) × G r ( R m ) and typical fibre GL r . The atlas of M r ( R n × 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 M r ( R n × m ) equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space R n × m equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space R n × m , seen as the union of manifolds M r ( R n × m ) , as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

Keywords: matrix manifolds; low-rank matrices; Grassmann manifold; principal bundles (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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