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Row or column completion of polynomial matrices of given degree II

Agurtzane Amparan, Itziar Baragaña, Silvia Marcaida, Alicia Roca

Abstract

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.

Row or column completion of polynomial matrices of given degree II

Abstract

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.
Paper Structure (7 sections, 12 theorems, 92 equations)

This paper contains 7 sections, 12 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Row (column) completion for polynomial matrices: remaining cases
  4. Prescription of infinite and singular structures
  5. Prescription of finite and singular structures
  6. Prescription of singular structures
  7. Conclusions

Key Result

Lemma 2.1

(Index Sum Theorem for Matrix Polynomials DeDoMa14) Let $P(s)\in\mathbb F[s]^{m\times n}$ be a polynomial matrix, $\mathop{\rm deg }\nolimits(P(s))=d$, $\mathop{\rm rank}\nolimits(P(s))=r$. Let $\phi_1(s,t)\mid\cdots\mid\phi_r(s,t)$, $c_1\geq\cdots\geq c_{n-r}$ and $u_1\geq\cdots\geq u_{m-r}$ be the

Theorems & Definitions (23)

  • Lemma 2.1
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 3.1
  • Remark 3.2
  • Example 3.3
  • Corollary 3.4
  • ...and 13 more