arXiv subjects

Agurtzane Amparan

Publications and source records attributed to Agurtzane Amparan.

4 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

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

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↗

arXiv2020

On minimal bases and indices of rational matrices and their linearizations

A complete theory of the relationship between the minimal bases and indices of rational matrices and those of their strong linearizations is presented. Such theory is based on establishing first the relationships between the minimal bases and indices of rational matrices and those of their polynomial system matrices under the classical minimality condition and certain additional conditions of properness. This is related to pioneer results obtained by Verghese, Van Dooren and Kailath in 1979-80, which were the first proving results of this type under different nonequivalent conditions. It is shown that the definitions of linearizations and strong linearizations do not guarantee any relationship between the minimal bases and indices of the linearizations and the rational matrices in general. In contrast, simple relationships are obtained for the family of strong block minimal bases linearizations, which can be used to compute minimal bases and indices of any rational matrix, including rectangular ones, via algorithms for pencils. These results extend the corresponding ones for other families of linearizations available in recent literature for square rational matrices.

math.NA↗