Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms
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 w...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2020 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/105930 |
| Acceso en línea: | http://hdl.handle.net/11336/105930 |
| Access Level: | acceso abierto |
| Palabra clave: | MINIMAL CURVES FINSLER METRICS UNITARY OPERATORS POSITIVE OPERATORS GRASSMANN MANIFOLD INTERMEDIATE POINTS https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | 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 Gl(n)+ endowed with the bi-invariant Finsler metric given by the trace norm, ‖X‖_1,A = ‖A^−1/2XA^−1/2‖_1 for any X tangent to A∈Gl(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. |
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