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 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

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 set of unattainable points for the Rational Hermite Interpolation Problem

We describe geometrically and algebraically the set of unattainable points for the Rational Hermite Interpolation Problem (i.e. those points where the problem does not have a solution). We show that this set is a union of equidimensional complete intersection varieties of odd codimension, the number of them being equal to the minimum between the degrees of the numerator and denominator of the problem. Each of these equidimensional varieties can be further decomposed as a union of as many rational (irreducible) varieties as input data points. We exhibit algorithms and equations defining all these objects

Bounds for degrees of syzygies of polynomials defining a grade two ideal

We make explicit the exponential bound on the degrees of the polynomials appearing in the Effective Quillen-Suslin Theorem, and apply it jointly with the Hilbert-Burch Theorem to show that the syzygy module of a sequence of $m$ polynomials in $n$ variables defining a complete intersection ideal of grade two is free, and that a basis of it can be computed with bounded degrees. In the known cases, these bounds improve previous results

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

Minimal solutions of the rational interpolation problem

We explore connections between the approach of solving the rational interpolation problem via resolutions of ideals and syzygies, and the standard method provided by the Extended Euclidean Algorithm (EEA). As a consequence, we obtain explicit descriptions for solutions of minimal degrees in terms of the degrees of elements appearing in the EEA. This result allows us to describe the minimal degree in a μ-basis of a polynomial planar parametrization in terms of a critical degree arising in the EEA

The set of unattainable points for the Rational Hermite Interpolation Problem

We describe geometrically and algebraically the set of unattainable points for the Rational Hermite Interpolation Problem (i.e. those points where the problem does not have a solution). We show that this set is a union of equidimensional complete intersection varieties of odd codimension, the number of them being equal to the minimum between the degrees of the numerator and denominator of the problem. Each of these equidimensional varieties can be further decomposed as a union of as many rational (irreducible) varieties as input data points. We exhibit algorithms and equations defining all these objects