The lattice of characteristic subspaces of an endomorphism with Jordan-Chevalley decomposition

Given an endomorphism A over a finite dimensional vector space having Jordan-Chevalley decomposition, the lattices of invariant and hyperinvariant subspaces of A can be obtained from the nilpotent part of this decomposition. We extend this result for lattices of characteristic subspaces. We also obtain a generalization of Shoda's Theorem about the characterization of the existence of characteristic non hyperinvariant subspaces

The lattice of characteristic subspaces of an endomorphism with Jordan-Chevalley decomposition

[EN] Given an endomorphism A over a finite dimensional vector space having Jordan-Chevalley decomposition, the lattices of invariant and hyperinvariant subspaces of A can be obtained from the nilpotent part of this decomposition. We extend this result for lattices of characteristic subspaces. We also obtain a generalization of Shoda's theorem about the characterization of the existence of characteristic non hyperinvariant subspaces. (C) 2018 Elsevier Inc. All rights reserved.The second author is partially supported by grant MTM2015-65361-P MINECO/FEDER, UE and MTM2017-90682-REDT. The third author is partially supported by grants MTM2017-83624-P and MTM2017-90682-REDT.Mingueza, D.; Montoro, ME.; Roca Martinez, A. (2018). The lattice of characteristic subspaces of an endomorphism with Jordan-Chevalley decomposition. Linear Algebra and its Applications. 558:63-73. https://doi.org/10.1016/j.laa.2018.08.005S637355

The characteristic subspace lattice of a linear transformation

[EN] Given a square matrix A in Mn(F), the lattices of the hyper-invariant (Hinv(A)) and characteristic (Chinv(A)) subspaces coincide whenever Fis not GF(2). If the characteristic polynomial of A splits over F, A can be considered nilpotent. In this paper we investigate the properties of the lattice Chinv(J) when F =GF(2) for a nilpotent matrix J. In particular, we prove it to be self-dual.The second author is partially supported by MINECO, grant MTM2015-65361-P and third author is partially supported by MINECO, grant MTM2013-40960-P, and by Gobierno Vasco, grant GIC13/IT-710-13.Mingueza, D.; Montoro, ME.; Roca Martinez, A. (2016). The characteristic subspace lattice of a linear transformation. Linear Algebra and its Applications. 506:329-341. https://doi.org/10.1016/j.laa.2016.06.003S32934150

Análisis de redes complejas: Aplicaciones en redes de colaboraciones científicas

Las redes complejas han sido ampliamente utilizadas en el estudio de redes de colaboración científica. En este trabajo se han utilizado métodos y magnitudes propios de las redes complejas y similaridad de conjuntos para la detección y análisis de comunidades en la red de colaboración científica del área de Ciencias de la Universidad de Zaragoz

Aplicativo de análisis automático de entrenamientos

Trainerer es una empresa que crea planes de entrenamiento para deportistas. Para ello disponen de una plataforma que importa datos de aplicaciones que registran entrenamientos como Strava y Garmin. Con estos datos se analizan los registros de las actividades y se guardan para crear un historial deportivo de cada atleta. Con el desarrollo de este proyecto se pretende dar solución al problema de analizar automáticamente el estado de forma de un deportista y predecir la evolución de sus parámetros de rendimiento a corto plazo. Para llevar a cabo esta tarea se ha utilizado el lenguaje de programación Python, juntamente con el framework Flask y el sistema de contenedores Docker.Trainerer is a company that makes training plans for athletes. To this end, they have a platform that imports data from external applications that register training like Strava and Garmin. The logs of the activities are analyzed with this data and they are saved to create a sporty track record for each athlete. With the development of this project, it is intended to provide a solution to the problem of analyzing automatically the condition of an athlete and predict the evolution of their performance parameters in a short term. To perform this task, the Python programming language has been used mainly, besides the Flask framework and the Docker container system.Trainerer és una empresa que crea plans d’entrenament per esportistes. Per això disposen d’una plataforma que importa dades d’aplicacions que registren entrenaments com Strava y Garmin. Amb aquestes dades s’analitzen els registres de les activitats i es guarden per crear un historial esportiu de cada atleta. Amb el desenvolupament d’aquest projecte es prenten proporcionar una solució al problema d’analitzar automàticament l’estat de forma d’un esportista i predir l’evolució dels seus paràmetres de rendiment en un període de temps curt. Per dur a terme aquesta tasca s’ha utilizat el llenguatge de programació Python, juntament amb el framework Flask i el sistema de contenidors Docker

Computing the cardinality of the lattice of characteristic subspaces

[EN] We obtain the cardinality of the lattice of characteristic sub-spaces of a nilpotent Jordan matrix when the underlying field is GF(2), the only field where the lattices of characteristic and hyperinvariant subspaces can be different. If the charac-teristic polynomial of the matrix splits in the field, the general case can be reduced to the nilpotent Jordan case. Results are complex and highly combinatorial, and include the design of an algorithm.The second author is partially supported by grant MTM2015-65361-P MINECO/FEDER, UE. The third author is partially supported by grants MTM2013-40960-P MINECO and MTM2015-68805-REDT.Mingueza, D.; Montoro, M.; Roca Martinez, A. (2017). Computing the cardinality of the lattice of characteristic subspaces. Linear Algebra and its Applications. 514:82-104. https://doi.org/10.1016/j.laa.2016.10.031S8210451

The centralizer of an endomorphism over an arbitrary field

[EN] The centralizer of an endomorphism of a finite dimensional vector space is known when the endomorphism is nonderogatory or when its minimal polynomial splits over the field. It is also known for the real Jordan canonical form. In this paper we characterize the centralizer of an endomorphism over an arbitrary field, and compute its dimension. The result is obtained via generalized Jordan canonical forms (for separable and nonseparable minimal polynomials). In addition, we also obtain the corresponding generalized Weyr canonical forms and the structure of its centralizers, which in turn allows us to compute the determinant of its elements. (C) 2020 Elsevier Inc. All rights reserved.The second author is partially supported by "Ministerio de Economía, Industria y Competitividad (MINECO)" of Spain and "Fondo Europeo de Desarrollo Regional (FEDER)" of EU through grants MTM2015-65361-P and MTM2017-90682-REDT. The third author is partially supported by "Ministerio de Economía, Industria y Competitividad (MINECO)" of Spain and "Fondo Europeo de Desarrollo Regional (FEDER)" of EU through grants MTM2017-83624-P and MTM2017-90682-REDT.Mingueza, D.; Montoro, ME.; Roca Martinez, A. (2020). The centralizer of an endomorphism over an arbitrary field. Linear Algebra and its Applications. 591:322-351. https://doi.org/10.1016/j.laa.2020.01.013S322351591Astuti, P., & Wimmer, H. K. (2009). Hyperinvariant, characteristic and marked subspaces. Operators and Matrices, (2), 261-270. doi:10.7153/oam-03-16Asaeda, Y. (1993). A remark to the paper «On the stabilizer of companion matrices» by J. Gomez-Calderon. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 69(6). doi:10.3792/pjaa.69.170Brickman, L., & Fillmore, P. A. (1967). The Invariant Subspace Lattice of a Linear Transformation. Canadian Journal of Mathematics, 19, 810-822. doi:10.4153/cjm-1967-075-4Dalalyan, S. H. (2014). Generalized Jordan Normal Forms of Linear Operators. Journal of Mathematical Sciences, 198(5), 498-504. doi:10.1007/s10958-014-1805-3Ferrer, J., Mingueza, D., & Montoro, M. E. (2013). Determinant of a matrix that commutes with a Jordan matrix. Linear Algebra and its Applications, 439(12), 3945-3954. doi:10.1016/j.laa.2013.10.023Gomez-Calderon, J. (1993). On the stabilizer of companion matrices. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 69(5). doi:10.3792/pjaa.69.140Fillmore, P. A., Herrero, D. A., & Longstaff, W. E. (1977). The hyperinvariant subspace lattice of a linear transformation. Linear Algebra and its Applications, 17(2), 125-132. doi:10.1016/0024-3795(77)90032-5Holtz, O. (2000). Applications of the duality method to generalizations of the Jordan canonical form. Linear Algebra and its Applications, 310(1-3), 11-17. doi:10.1016/s0024-3795(00)00054-9Mingueza, D., Eulàlia Montoro, M., & Pacha, J. R. (2013). Description of characteristic non-hyperinvariant subspaces over the fieldGF(2). Linear Algebra and its Applications, 439(12), 3734-3745. doi:10.1016/j.laa.2013.10.025Mingueza, D., Montoro, M. E., & Roca, A. (2018). The lattice of characteristic subspaces of an endomorphism with Jordan–Chevalley decomposition. Linear Algebra and its Applications, 558, 63-73. doi:10.1016/j.laa.2018.08.005Robinson, D. W. (1965). On Matrix Commutators of Higher Order. Canadian Journal of Mathematics, 17, 527-532. doi:10.4153/cjm-1965-052-9Robinson, D. W. (1970). The Generalized Jordan Canonical Form. The American Mathematical Monthly, 77(4), 392-395. doi:10.1080/00029890.1970.1199250

Description of characteristic non-hyperinvariant subspaces in GF(2)

Given a square matrix A , an A -invariant subspace is called hyperinvariant (respectively, characteristic) if and only if it is also invariant for all matrices T (respectively, nonsingular matrices T ) that commute with A . Shoda's Theorem gives a necessary and sufficient condition for the existence of characteristic non-hyperinvariant subspaces for a nilpotent matrix in GF(2)GF(2). Here we present an explicit construction for all subspaces of this type.Peer ReviewedPostprint (published version

Description of characteristic non-hyperinvariant subspaces in GF(2)

Given a square matrix A , an A -invariant subspace is called hyperinvariant (respectively, characteristic) if and only if it is also invariant for all matrices T (respectively, nonsingular matrices T ) that commute with A . Shoda's Theorem gives a necessary and sufficient condition for the existence of characteristic non-hyperinvariant subspaces for a nilpotent matrix in GF(2)GF(2). Here we present an explicit construction for all subspaces of this type.Peer Reviewe

Description of characteristic non-hyperinvariant subspaces over the field GF(2)

Given a square matrix A, an A-invariant subspace is called hyperinvariant (respectively, characteristic) if and only if it is also invariant for all matrices T (respectively, nonsingular matrices T) that commute with A. Shodaʼs Theorem gives a necessary and sufficient condition for the existence of characteristic non-hyperinvariant subspaces for a nilpotent matrix in GF(2). Here we present an explicit construction for all subspaces of this type.Peer Reviewe