Principal Bundle Structure of Matrix Manifolds
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 <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="d...
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MDPI AG
2021-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/14/1669 |
| _version_ | 1797526646794747904 |
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| author | Marie Billaud-Friess Antonio Falcó Anthony Nouy |
| author_facet | Marie Billaud-Friess Antonio Falcó Anthony Nouy |
| author_sort | Marie Billaud-Friess |
| collection | DOAJ |
| description | 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 <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">G</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>k</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of linear subspaces of dimension <inline-formula><math display="inline"><semantics><mrow><mi>r</mi><mo><</mo><mi>k</mi></mrow></semantics></math></inline-formula> in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>k</mi></msup></semantics></math></inline-formula>, which avoids the use of equivalence classes. The set <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">G</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>k</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mi>r</mi><mo>)</mo><mo>×</mo><mi>r</mi></mrow></msup></semantics></math></inline-formula>. Then, we define an atlas for the set <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>k</mi><mo>×</mo><mi>r</mi></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">G</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>k</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and typical fibre <inline-formula><math display="inline"><semantics><msub><mi>GL</mi><mi>r</mi></msub></semantics></math></inline-formula>, the general linear group of invertible matrices in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mi>k</mi><mo>×</mo><mi>k</mi></mrow></msup></semantics></math></inline-formula>. Finally, we define an atlas for the set <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">G</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>)</mo></mrow><mo>×</mo><msub><mi mathvariant="double-struck">G</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>m</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and typical fibre <inline-formula><math display="inline"><semantics><msub><mi>GL</mi><mi>r</mi></msub></semantics></math></inline-formula>. The atlas of <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> 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 <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup></semantics></math></inline-formula> equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup></semantics></math></inline-formula>, seen as the union of manifolds <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, as an analytic manifold equipped with a topology for which the matrix rank is a continuous map. |
| first_indexed | 2024-03-10T09:32:16Z |
| format | Article |
| id | doaj.art-44a9451c4f1a45ad9a726fa60e68d3b3 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T09:32:16Z |
| publishDate | 2021-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-44a9451c4f1a45ad9a726fa60e68d3b32023-11-22T04:20:22ZengMDPI AGMathematics2227-73902021-07-01914166910.3390/math9141669Principal Bundle Structure of Matrix ManifoldsMarie Billaud-Friess0Antonio Falcó1Anthony Nouy2Department of Computer Science and Mathematics, Ecole Centrale de Nantes, 1 Rue de la Noë, BP 92101, CEDEX 3, 44321 Nantes, FranceDepartamento de Matemáticas, Física y Ciencias Tecnológicas, Universidad CEU Cardenal Herrera, CEU Universities, San Bartolomé 55, 46115 Alfara del Patriarca, SpainDepartment of Computer Science and Mathematics, Ecole Centrale de Nantes, 1 Rue de la Noë, BP 92101, CEDEX 3, 44321 Nantes, FranceIn 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 <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">G</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>k</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of linear subspaces of dimension <inline-formula><math display="inline"><semantics><mrow><mi>r</mi><mo><</mo><mi>k</mi></mrow></semantics></math></inline-formula> in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>k</mi></msup></semantics></math></inline-formula>, which avoids the use of equivalence classes. The set <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">G</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>k</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mi>r</mi><mo>)</mo><mo>×</mo><mi>r</mi></mrow></msup></semantics></math></inline-formula>. Then, we define an atlas for the set <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>k</mi><mo>×</mo><mi>r</mi></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">G</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>k</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and typical fibre <inline-formula><math display="inline"><semantics><msub><mi>GL</mi><mi>r</mi></msub></semantics></math></inline-formula>, the general linear group of invertible matrices in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mi>k</mi><mo>×</mo><mi>k</mi></mrow></msup></semantics></math></inline-formula>. Finally, we define an atlas for the set <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">G</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>)</mo></mrow><mo>×</mo><msub><mi mathvariant="double-struck">G</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>m</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and typical fibre <inline-formula><math display="inline"><semantics><msub><mi>GL</mi><mi>r</mi></msub></semantics></math></inline-formula>. The atlas of <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> 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 <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup></semantics></math></inline-formula> equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup></semantics></math></inline-formula>, seen as the union of manifolds <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mi>r</mi></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.https://www.mdpi.com/2227-7390/9/14/1669matrix manifoldslow-rank matricesGrassmann manifoldprincipal bundles |
| spellingShingle | Marie Billaud-Friess Antonio Falcó Anthony Nouy Principal Bundle Structure of Matrix Manifolds Mathematics matrix manifolds low-rank matrices Grassmann manifold principal bundles |
| title | Principal Bundle Structure of Matrix Manifolds |
| title_full | Principal Bundle Structure of Matrix Manifolds |
| title_fullStr | Principal Bundle Structure of Matrix Manifolds |
| title_full_unstemmed | Principal Bundle Structure of Matrix Manifolds |
| title_short | Principal Bundle Structure of Matrix Manifolds |
| title_sort | principal bundle structure of matrix manifolds |
| topic | matrix manifolds low-rank matrices Grassmann manifold principal bundles |
| url | https://www.mdpi.com/2227-7390/9/14/1669 |
| work_keys_str_mv | AT mariebillaudfriess principalbundlestructureofmatrixmanifolds AT antoniofalco principalbundlestructureofmatrixmanifolds AT anthonynouy principalbundlestructureofmatrixmanifolds |