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A Perturbation Method for Index Detection for Linear Matrix Pencils

Hanna Blazhko, Michał Wojtylak

Abstract

Rigorous, non-asymptotic bounds for the Puiseux expansion of the eigenvalue at infinity are given. Error analysis is provided. Further, the expected value of the eigenvector condition number of a randomly perturbed matrix is estimated. The latter result is applied to the Cayley transform of the linear pencil. Numerical simulations illustrating the theoretical findings are provided.

A Perturbation Method for Index Detection for Linear Matrix Pencils

Abstract

Rigorous, non-asymptotic bounds for the Puiseux expansion of the eigenvalue at infinity are given. Error analysis is provided. Further, the expected value of the eigenvector condition number of a randomly perturbed matrix is estimated. The latter result is applied to the Cayley transform of the linear pencil. Numerical simulations illustrating the theoretical findings are provided.
Paper Structure (15 sections, 7 theorems, 49 equations, 2 figures)

This paper contains 15 sections, 7 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Deterministic results
  3. Probabilistic results
  4. Organization of the paper
  5. Theorem \ref{['Theorem1']}: auxiliary results, proof, and corollaries
  6. Port-Hamiltonian pencils
  7. Proof of Theorem \ref{['thmM']}
  8. Practical implementations
  9. Toy model
  10. The influence of a congruence transformation
  11. A dissipative Hamiltonian linearisation of a system of vibrating strings
  12. Electrical circuit
  13. Errors in the staircase form
  14. Conclusions
  15. Acknowledgments

Key Result

Theorem 1.1

Let $\lambda E -A\in\mathbb C^{n,n}[\lambda]$ be a regular pencil of index two with no Kronecker blocks corresponding to infinity of size one and with $E\geq 0$. Then a perturbed pencil is regular and has an eigenvalue $\mu_\Delta(\tau)$ satisfying where

Figures (2)

  • Figure 1: The plots show the reciprocals of eigenvalues corresponding to Theorem \ref{['Theorem1']} (left column, solid blue line) and the eigenvalues corresponding to Theorem \ref{['thmM']} (right column, solid blue line). The corresponding bounds are shown with dash-dot orange lines. See Remarks \ref{['rem:leftplot']} and \ref{['rem:rightplot']} for a detailed description. The rows correspond to the subsections, as indicated.
  • Figure 2: The plots show the reciprocals of eigenvalues corresponding to Theorem \ref{['Theorem1']} with solid blue lines. Corresponding bounds are shown with dash-dot orange lines. See Remark \ref{['rem:leftplot']} for a full description. The plots correspond to the variance $e^{-7}$, $e^{-5}$, $e^{-3}$ of the entries of $\Delta_A$ from Subsection \ref{['ex:deltas']}, respectively.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • proof : Proof of Theorem \ref{['Theorem1']}
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 10 more