Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms
Abstract Consider the Lie group of n × n complex unitary matrices U ( n ) endowed with the bi-invariant Finsler metric given by the spectral norm, ‖ X ‖ U = ‖ U ⁎ X ‖ ∞ = ‖ X ‖ ∞ for any X tangent to a unitary operator U. Given two points in U ( n ) , in general there exist infinitely many curves of...
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| Autores principales: | , , |
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| Formato: | Articulo Preprint |
| Lenguaje: | Inglés |
| Publicado: |
2020
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| Acceso en línea: | http://sedici.unlp.edu.ar/handle/10915/126676 |
| Aporte de: |
| id |
I19-R120-10915-126676 |
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| record_format |
dspace |
| institution |
Universidad Nacional de La Plata |
| institution_str |
I-19 |
| repository_str |
R-120 |
| collection |
SEDICI (UNLP) |
| language |
Inglés |
| topic |
Matemática Combinatorics Grassmannian Lie group Unitary operator Trace (linear algebra) Mathematics Matrix norm Hermitian matrix Unitary matrix |
| spellingShingle |
Matemática Combinatorics Grassmannian Lie group Unitary operator Trace (linear algebra) Mathematics Matrix norm Hermitian matrix Unitary matrix Antezana, Jorge Abel Ghiglioni, Eduardo Mario Stojanoff, Demetrio Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms |
| topic_facet |
Matemática Combinatorics Grassmannian Lie group Unitary operator Trace (linear algebra) Mathematics Matrix norm Hermitian matrix Unitary matrix |
| description |
Abstract Consider the Lie group of n × n complex unitary matrices U ( n ) endowed with the bi-invariant Finsler metric given by the spectral norm, ‖ X ‖ U = ‖ U ⁎ X ‖ ∞ = ‖ X ‖ ∞ for any X tangent to a unitary operator U. Given two points in U ( n ) , in general there exist infinitely many curves of minimal length. In this paper we provide a complete description of such curves and as a consequence we give an equivalent condition for uniqueness. Similar studies are done for the Grassmann manifolds. On the other hand, consider the cone of n × n positive invertible matrices G l ( n ) + endowed with the bi-invariant Finsler metric given by the trace norm, ‖ X ‖ 1 , A = ‖ A − 1 / 2 X A − 1 / 2 ‖ 1 for any X tangent to A ∈ G l ( n ) + . In this context, also exist infinitely many curves of minimal length. In this paper we provide a complete description of such curves proving first a characterization of the minimal curves joining two Hermitian matrices X , Y ∈ H ( n ) . The last description is also used to construct minimal paths in the group of unitary matrices U ( n ) endowed with the bi-invariant Finsler metric ‖ X ‖ 1 , U = ‖ U ⁎ X ‖ 1 = ‖ X ‖ 1 for any X tangent to U ∈ U ( n ) . We also study the set of intermediate points in all the previous contexts. |
| format |
Articulo Preprint |
| author |
Antezana, Jorge Abel Ghiglioni, Eduardo Mario Stojanoff, Demetrio |
| author_facet |
Antezana, Jorge Abel Ghiglioni, Eduardo Mario Stojanoff, Demetrio |
| author_sort |
Antezana, Jorge Abel |
| title |
Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms |
| title_short |
Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms |
| title_full |
Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms |
| title_fullStr |
Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms |
| title_full_unstemmed |
Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms |
| title_sort |
minimal curves in u(n) and gl(n)+ with respect to the spectral and the trace norms |
| publishDate |
2020 |
| url |
http://sedici.unlp.edu.ar/handle/10915/126676 |
| work_keys_str_mv |
AT antezanajorgeabel minimalcurvesinunandglnwithrespecttothespectralandthetracenorms AT ghiglionieduardomario minimalcurvesinunandglnwithrespecttothespectralandthetracenorms AT stojanoffdemetrio minimalcurvesinunandglnwithrespecttothespectralandthetracenorms |
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Repositorios |
| _version_ |
1764820450993504257 |