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Luís Carvalho,
Cristina Diogo, and
Sérgio Mendes
On the convexity and circularity of the numerical range of nilpotent quaternionic matrices
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| Published: |
December 15, 2019. |
| Keywords: |
quaternions, numerical range, nilpotent matrix. |
| Subject: |
15B33, 47A12. |
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Abstract
We provide a sufficient condition for the numerical range of a nilpotent matrix N to be circular in terms of the existence of cycles in an undirected graph associated with N. We prove that if we add to this matrix N a diagonal real matrix D, the matrix D+N has convex numerical range. For 3 x 3 nilpotent matrices, we strength further our results and obtain necessary and sufficient conditions for circularity and convexity of the numerical range. |
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| Acknowledgements
The second author was partially supported by FCT through project UID/MAT/04459/2013 and the third author was partially supported by FCT through CMA-UBI,
project PEst-OE/MAT/UI0212/2013.
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| Author information
Luís Carvalho:
ISCTE - Lisbon University Institute
Av. das Forças Armadas
1649-026 Lisbon, Portugal
luis.carvalho@iscte-iul.pt
Cristina Diogo:
ISCTE - Lisbon University Institute
Av. das Forças Armadas
1649-026 Lisbon, Portugal, and
Center for Mathematical Analysis, Geometry, and Dynamical Systems
Mathematics Department, Instituto Superior Técnico
Universidade de Lisboa
Av. Rovisco Pais, 1049-001 Lisboa, Portugal
cristina.diogo@iscte-iul.pt
Sérgio Mendes:
ISCTE - Lisbon University Institute
Av. das Forças Armadas
1649-026 Lisbon, Portugal, and
Centro de Matemática e Aplicações
Universidade da Beira Interior
Rua Marquês d'Ávila e Bolama
6201-001 Covilhã, Portugal
sergio.mendes@iscte-iul.pt
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