arXiv subjects

Alicia Roca

Publications and source records attributed to Alicia Roca.

11 recordsLinked to original sources

arXiv2025

Row completion of polynomial and rational matrices II

We study the row completion problem of polynomial and rational matrices with partial prescription of the structural data. The prescription of the complete structural data has been solved in Amparan et al., Lin. Alg. Appl. 720 (2025) 109-138, where several results of prescription of some of the four types of invariants composing the structural data have also been obtained. In this paper we deal with the cases not analyzed there. More precisely, we solve the row completion problem of a rational or a polynomial matrix when we prescribe the infinite (finite) structure and the column and/or the row minimal indices, and when only the column and/or row minimal indices are prescribed. The necessity of the conditions obtained are valid over arbitrary fields, but in some cases the proof of the sufficiency requires working over algebraically closed fields. By transposition the results obtained hold for the corresponding column completion problems.

math.GM↗

arXiv2025

Bounds for the change of the Weyr characteristic of matrix pencils after 1-rank perturbations

The complete characterization of the Kronecker structure of a matrix pencil perturbed by another pencil of rank one is known, and it is stated in terms of very involved conditions. This paper is devoted to, loosing accuracy, better understand the meaning of those conditions. The Kronecker structure of a pencil is determined by the sequences of the column and row minimal indices and of the partial multiplicities of the eigenvalues. We introduce the Weyr characteristic of a matrix pencil as the collection of the conjugate partitions of the previous sequences and provide bounds for the change of each one of the Weyr partitions when the pencil is perturbed by a pencil of rank one. For each one of the Weyr components, the resulting bounds are expressed only in terms of the corresponding component of the unperturbed pencil. In order to verify that the bounds are reachable, we also characterize the partitions of the Weyr characteristic of a pencil obtained from another one by removing or adding one row. The results hold for algebraically closed fields

math.AC↗

arXiv2025

Polynomial and rational matrices with the invariant rational functions and the four sequences of minimal indices prescribed

The complete eigenstructure, or structural data, of a rational matrix $R(s)$ is comprised by its invariant rational functions, both finite and at infinity, which in turn determine its finite and infinite pole and zero structures, respectively, and by the minimal indices of its left and right null spaces. These quantities arise in many applications and have been thoroughly studied in numerous references. However, other two fundamental subspaces of $R(s)$ in contrast have received much less attention: the column and row spaces, which also have their associated minimal indices. This work solves the problems of finding necessary and sufficient conditions for the existence of rational matrices in two scenarios: (a) when the invariant rational functions and the minimal indices of the column and row spaces are prescribed, and (b) when the complete eigenstructure together with the minimal indices of the column and row spaces are prescribed. The particular, but extremely important, cases of these problems for polynomial matrices are solved first and are the main tool for solving the general problems. The results in this work complete and non-trivially extend the necessary and sufficient conditions recently obtained for the existence of polynomial and rational matrices when only the complete eigenstructure is prescribed.

math.SP↗

arXiv2025

Row completion of polynomial and rational matrices

We characterize the existence of a polynomial (rational) matrix when its eigenstructure (complete structural data) and some of its rows are prescribed. For polynomial matrices, this problem was solved in a previous work when the polynomial matrix has the same degree as the prescribed submatrix. In that paper, the following row completion problems were also solved arising when the eigenstructure was partially prescribed, keeping the restriction on the degree: the eigenstructure but the row (column) minimal indices, and the finite and/or infinite structures. Here we remove the restriction on the degree, allowing it to be greater than or equal to that of the submatrix. We also generalize the results to rational matrices. Obviously, the results obtained hold for the corresponding column completion problems.

math.SP↗

arXiv2024

Isomorphisms between lattices of hyperinvariant subspaces

Given two nilpotent endomorphisms, we determine when their lattices of hyperinvariant subspaces are isomorphic. The study of the lattice of hyperinvariant subspaces can be reduced to the nilpotent case when the endomorphism has a Jordan-Chevalley decomposition; for example, it occurs if the underlying field is the field of complex numbers.

math.RA↗

arXiv2024

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.

math.RA↗

arXiv2024

On the change of the Weyr characteristics of matrix pencils after rank-one perturbations

The change of the Kronecker structure of a matrix pencil perturbed by another pencil of rank one has been characterized in terms of the homogeneous invariant factors and the chains of column and row minimal indices of the initial and the perturbed pencils. We obtain here a new characterization in terms of the homogeneous invariant factors and the conjugate partitions of the corresponding chains of column and row minimal indices of both pencils. We also define the generalized Weyr characteristic of an arbitrary matrix pencil and obtain bounds for the change of it when the pencil is perturbed by another pencil of rank one. The results improve known results on the problem, hold for arbitrary perturbation pencils of rank one, and for any algebraically closed field.

math.RA↗

arXiv2024

Row or column completion of polynomial matrices of given degree II

The row (column) completion problem of polynomial matrices of given degree with prescribed eigenstructure has been studied in \cite{AmBaMaRo23}, where several results of prescription of some of the four types of invariants that form the eigenstructure have also been obtained. In this paper we conclude the study, solving the completion for the cases not covered there. More precisely, the row completion problem of a polynomial matrix is solved when we prescribe the infinite (finite) structure and column and/or row minimal indices, and finally the column and/or row minimal indices. The necessity of the characterizations obtained holds to be true over arbitrary fields in all cases, whilst to prove the sufficiency it is required, in some of the cases, to work over algebraically closed fields.

math.RA↗

arXiv2019

Rank-one perturbations of matrix pencils

We solve the problem of characterizing the Kronecker structure of a matrix pencil obtained by a rank-one perturbation of another matrix pencil. The results hold over arbitrary fields.

math.CO↗

arXiv2019

Fixed rank perturbations of regular matrix pencils

A characterization of the structure of a regular matrix pencil obtained by a bounded rank perturbation of another regular matrix pencil has been recently obtained. The result generalizes the solution for the bounded rank perturbation problem of a square constant matrix. When comparing the fixed rank perturbation problem of a constant matrix with the bounded rank perturbation problem we realize that both problems are of different nature; the first one is more restrictive. In this paper we characterize the structure of a regular matrix pencil obtained by a fixed rank perturbation of another regular matrix pencil. We apply the result to find necessary and sufficient conditions for the existence of a fixed rank perturbation such that the perturbed pencil has a prescribed determinant. The results hold over fields with sufficient number of elements.

math.AG↗

arXiv2018

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.

math.RA↗