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Row or column completion of polynomial matrices of given degree

A. Amparan, I. Baragaña, S. Marcaida, A. Roca

Abstract

We solve the problem of characterizing the existence of a polynomial matrix of fixed degree when its eigenstructure (or part of it) and some of its rows (columns) are prescribed. More specifically, we present a solution to the row (column) completion problem of a polynomial matrix of given degree under different prescribed invariants: the whole eigenstructure, all of it but the row (column) minimal indices, and the finite and/or infinite structures. Moreover, we characterize the existence of a polynomial matrix with prescribed degree and eigenstructure over an arbitrary field.

Row or column completion of polynomial matrices of given degree

Abstract

We solve the problem of characterizing the existence of a polynomial matrix of fixed degree when its eigenstructure (or part of it) and some of its rows (columns) are prescribed. More specifically, we present a solution to the row (column) completion problem of a polynomial matrix of given degree under different prescribed invariants: the whole eigenstructure, all of it but the row (column) minimal indices, and the finite and/or infinite structures. Moreover, we characterize the existence of a polynomial matrix with prescribed degree and eigenstructure over an arbitrary field.
Paper Structure (12 sections, 15 theorems, 76 equations)

This paper contains 12 sections, 15 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and previous results
  4. Row completion of matrix pencils
  5. Frobenius companion form
  6. Polynomial matrices with prescribed eigenstructure
  7. Row (column) completion of polynomial matrices of given degree
  8. Prescription of the whole eigenstructure
  9. Prescription of finite and infinite structures and column minimal indices
  10. Prescription of finite and infinite structures and row minimal indices
  11. Prescription of finite and/or infinite structures
  12. Conclusions and future work

Key Result

Lemma 2.3

Let $P(s)\in\mathbb F[s]^{m\times n}$, $\mathop{\rm deg }\nolimits(P(s))=d$, $\mathop{\rm rank}\nolimits(P(s))=r$. Let $\gamma_1(s,t)\mid\cdots\mid\gamma_r(s,t)$, $d_1\geq\cdots\geq d_{n-r}$ and $v_1\geq\cdots\geq v_{m-r}$ be the homogeneous invariant factors, column minimal indices and row minimal

Theorems & Definitions (24)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3: Index Sum Theorem for Matrix Polynomials DeDoMa14
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Definition 2.8
  • Lemma 2.9
  • Corollary 2.10
  • ...and 14 more