Row completion of polynomial and rational matrices
Agurtzane Amparan, Itziar Baragaña, Silvia Marcaida, Alicia Roca
TL;DR
This paper advances the matrix completion problem to polynomial and rational matrices by removing the degree-equality restriction on the completed matrix and by treating the rational case alongside the polynomial case. It frames the problem in terms of complete structural data, including invariant factors, infinite-structure data, and minimal indices, and provides explicit necessary-and-sufficient conditions for the existence of completions that realize prescribed invariants. The core methodology connects row/column completion to unimodular embeddings of invariant factors and to matrix pencils via the Frobenius companion form, complemented by generalized majorization conditions that govern the feasible changes in finite and infinite structures. The results unify polynomial and rational cases, extend prior degree-restricted findings, and lay out constructive criteria that can guide applications in control and systems theory where prescribed spectral and singular-structure properties must be realized through augmentation.
Abstract
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.