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Weierstrass structure and eigenvalue placement of regular matrix pencils under low rank perturbations

Itziar Baragaña, Alicia Roca

TL;DR

The problem of determining the Weierstrass structure of a regular matrix pencil obtained by a low rank perturbation of anotherregular matrix pencil is solved.

Abstract

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.

Weierstrass structure and eigenvalue placement of regular matrix pencils under low rank perturbations

TL;DR

The problem of determining the Weierstrass structure of a regular matrix pencil obtained by a low rank perturbation of anotherregular matrix pencil is solved.

Abstract

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.
Paper Structure (8 sections, 20 theorems, 67 equations)

This paper contains 8 sections, 20 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Statement of the problems
  4. Low rank perturbation for regular matrix pencils
  5. Necessary conditions
  6. Sufficiency of the conditions
  7. Eigenvalue placement for regular matrix pencils under low rank perturbations
  8. Conclusions

Key Result

Theorem 2.1

Two regular matrix pencils are strictly equivalent if and only if they have the same homogeneous invariant factors.

Theorems & Definitions (27)

  • Theorem 2.1: Weierstrass
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 4.1: Sa79
  • Remark 4.2
  • Proposition 4.3
  • Remark 4.4
  • Remark 4.5
  • ...and 17 more