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On the change of the Weyr characteristics of matrix pencils after rank-one perturbations

Itziar Baragaña, Alicia Roca

Abstract

The change of the Kronecker structure of a matrix pencil perturbed by another pencil of rank one has been characterized in terms of the homogeneous invariant factors and the chains of column and row minimal indices of the initial and the perturbed pencils. We obtain here a new characterization in terms of the homogeneous invariant factors and the conjugate partitions of the corresponding chains of column and row minimal indices of both pencils. We also define the generalized Weyr characteristic of an arbitrary matrix pencil and obtain bounds for the change of it when the pencil is perturbed by another pencil of rank one. The results improve known results on the problem, hold for arbitrary perturbation pencils of rank one, and for any algebraically closed field.

On the change of the Weyr characteristics of matrix pencils after rank-one perturbations

Abstract

The change of the Kronecker structure of a matrix pencil perturbed by another pencil of rank one has been characterized in terms of the homogeneous invariant factors and the chains of column and row minimal indices of the initial and the perturbed pencils. We obtain here a new characterization in terms of the homogeneous invariant factors and the conjugate partitions of the corresponding chains of column and row minimal indices of both pencils. We also define the generalized Weyr characteristic of an arbitrary matrix pencil and obtain bounds for the change of it when the pencil is perturbed by another pencil of rank one. The results improve known results on the problem, hold for arbitrary perturbation pencils of rank one, and for any algebraically closed field.

Paper Structure

This paper contains 8 sections, 17 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Jordan chains of matrix pencils
  4. Theorem \ref{['maintheogenpenc']} in terms of the conjugate partitions
  5. Bounds
  6. Technical Lemmas
  7. Main theorem
  8. Conclusion

Key Result

Theorem 2.2

Let $A(s), B(s)\in \mathbb F[s]^{p\times q}$ be matrix pencils such that $A(s)\centernot \stackrel{s.e.}{\sim} B(s)$. Let $\mathop{\rm rank}\nolimits A(s)=\rho_1$, $\mathop{\rm rank}\nolimits B(s)=\rho_2$, let $\phi_1(s, t)\mid \dots \mid \phi_{\rho_1}(s, t)$, $c_1 \geq \dots \geq c_{q-\rho_1}\geq

Theorems & Definitions (23)

  • Definition 2.1: $1$step-generalized majorization
  • Theorem 2.2: BaRo20_2
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Corollary 3.6
  • Theorem 3.7
  • Remark 3.8
  • ...and 13 more