Weierstrass structure and eigenvalue placement of regular matrix pencils under low rank perturbations

We solve the problem of determining the Weierstrass structure of a regular matrix pencil obtained by a low rank perturbation of another regular matrix pencil. We apply the result to find necessary and sufficient conditions for the existence of a low rank perturbation such that the perturbed pencil has prescribed eigenvalues and algebraic multiplicities. The results hold over fields with sufficient number of elements.Comment: 17 pages. arXiv admin note: text overlap with arXiv:1907.1065

Muting gender in the borderlands: still a bio-geopolitical open wound

Este ensayo es el resultado del estudio del silenciamientode la voz de la mujer, como parte de la tradición de un discursorepresentativo de la identidad mejicana, que tiene su raíz en la necesidad de reconocer su historia y un pasado de dolor, que esperan ser reparados y analizados. En este sentido, la feminización de un pueblo derrotado justificado por Doña Marina como la lengua de la conquista, deja una impronta psicológica bajo la cual el mejicano se siente perdido y abandonado por la Madre que ha dado a luz al primer mestizo mejicano. En esta intersección de la pérdida de una nación, el abandono de la madre y el odio hacia sí mismo, “the loss of a sense of dignity and respect in the macho breeds a false machismo which leads him to put down women and even to brutalize them. Coexisting with his sexist behaviour is a love for the mother which takes precedence over that of all others” (Anzandúa 1987c, 83). La reconciliación con esta vergonzosa ancestralidad llega de la mano de la figuración sincrética de la Virgen de Guadalupe: diosa mestiza cristiana legada al pueblo mejicano. En este trabajo, enlazaré la relación existente entre la identidad nacional mejicana y su representación iconográfica entre mujer/nación. De igual manera, trazaré un puente entre la historia no reconocida del pasado mejicano y el efecto psicológico de la conquista española y con asuntos relacionados con la frontera entre Méjico y los Estados Unidos de América, lo cual incide en una doble conquista y sus consecuentes negociaciones en la construcción de la subjetividad mejicana.Palabras clave: Malinche, Virgen de Guadalupe, identidad, materia(rea)lidad del cuerpo, la chingada, voz femenina, narrativa nacional.Abstract In this essay I will explore how silencing of women’s voice has a tradition conforming a discourse of the Mexican identity, that comes from their unacknowledged history, their not coming to terms with a hurting past that needs to be addressed and foreclosed. In this instance, the feminization of a defeated people justified by Doña Marina as the tongue of the conquest, leaves a psychological imprint in which the Mexican feels at a loss and abandoned by the Mother who has born the first mestizo Mexican son. At this intersection of loss of a nation, abandonment by the mother and self hate, “the loss of a sense of dignity and respect in the macho breeds a false machismo which leads him to put down women and even to brutalize them. Coexisting with his sexist behaviour is a love for the mother which takes precedence over that of all others” (Anzandúa 1987c, 83). Reconciliation with that shameful ancestrality comes with the syncretic figuration of the Virgin of Guadalupe, a mestiza Christian given goddess. Linking this national identity and iconographic representation of women/nation, I will also make a bridge between this unacknowledged history and the psychological effect of the Spanish conquest and issues dealing with the borderline between Mexico and the USA, which makes for a double conquest and the subsequent negotiations of subjectivities.Key words: Malinche, Virgen de Guadalupe; identity; body matterreality; la chingada; female voice; national narrative

Linear feedback shift registers and the minimal realization problem

[EN] The Berlekamp-Massey algorithm solves the problem of finding the shortest linear feedback shift register which generates a given finite sequence of scalars. This problem is reinterpreted from the point of view of the realization theory and several extensions to sequences of matrices are analyzed. We give a generalization of the result on which the Berlekamp-Massey algorithm is based in terms of the partial Brunovsky indices of a sequence of matrices and propose an algorithm to obtain them for sequences of vectors. The results we obtain hold for arbitrary fields.The first author is partially supported by grants MINECO MTM2017-83624-P, MTM2017-90682-REDT, and UPV/EHU GIU16/42. The second author is partially supported by grants MINECO MTM2017-83624-P and MTM2017-90682-REDT.Baragana, I.; Roca Martinez, A. (2019). Linear feedback shift registers and the minimal realization problem. Linear Algebra and its Applications. 576:200-227. https://doi.org/10.1016/j.laa.2018.06.009S20022757

Weierstrass Structure and Eigenvalue Placement of Regular Matrix Pencils under Low Rank Perturbations

[EN] We solve the problem of determining the Weierstrass structure of a regular matrix pencil obtained by a low rank perturbation of another regular matrix pencil. We apply the result to find necessary and sufficient conditions for the existence of a low rank perturbation such that the perturbed pencil has prescribed eigenvalues and algebraic multiplicities. The results hold over fields with sufficient number of elements.The work of the first and second authors was partially supported by MINECO grants MTM2017-83624-P and MTM2017-90682-REDT. The work of the first author was also partially supported by UPV/EHU grant GIU16/42.Baragana Garate, I.; Roca Martinez, A. (2019). Weierstrass Structure and Eigenvalue Placement of Regular Matrix Pencils under Low Rank Perturbations. SIAM Journal on Matrix Analysis and Applications. 40(2):440-453. https://doi.org/10.1137/18M120024544045340

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