Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Solving singular generalized eigenvalue problems. Part III: structure preservation

Michiel E. Hochstenbach, Christian Mehl, Bor Plestenjak

TL;DR

The paper addresses singular real symmetric or complex Hermitian generalized eigenvalue problems by developing structure-preserving methods that maintain the sign characteristic of the pencil. It introduces Hermitian rank-completing perturbations and a structure-preserving projection approach, proving that the perturbed or projected pencils separate eigenvalues into true, prescribed, and random categories, with random eigenvalues generically nonreal and simple. A key contribution is the preservation of the sign characteristic for true eigenvalues and the ability to recover it from perturbed pencils, enabling reliable analysis of invariant properties during computation. The results extend to other symmetry structures via transformations and demonstrate practical applications to bivariate polynomial systems using symmetric determinantal representations, supported by numerical experiments on illustrative pencils.

Abstract

In Parts I and II of this series of papers, three new methods for the computation of eigenvalues of singular pencils were developed: rank-completing perturbations, rank-projections, and augmentation. It was observed that a straightforward structure-preserving adaption for symmetric pencils was not possible and it was left as an open question how to address this challenge. In this Part III, it is shown how the observed issue can be circumvented by using Hermitian perturbations. This leads to structure-preserving analogues of the three techniques from Parts I and II for Hermitian pencils (including real symmetric pencils) as well as for related structures. It is an important feature of these methods that the sign characteristic of the given pencil is preserved. As an application, it is shown that the resulting methods can be used to solve systems of bivariate polynomials.

Solving singular generalized eigenvalue problems. Part III: structure preservation

TL;DR

The paper addresses singular real symmetric or complex Hermitian generalized eigenvalue problems by developing structure-preserving methods that maintain the sign characteristic of the pencil. It introduces Hermitian rank-completing perturbations and a structure-preserving projection approach, proving that the perturbed or projected pencils separate eigenvalues into true, prescribed, and random categories, with random eigenvalues generically nonreal and simple. A key contribution is the preservation of the sign characteristic for true eigenvalues and the ability to recover it from perturbed pencils, enabling reliable analysis of invariant properties during computation. The results extend to other symmetry structures via transformations and demonstrate practical applications to bivariate polynomial systems using symmetric determinantal representations, supported by numerical experiments on illustrative pencils.

Abstract

In Parts I and II of this series of papers, three new methods for the computation of eigenvalues of singular pencils were developed: rank-completing perturbations, rank-projections, and augmentation. It was observed that a straightforward structure-preserving adaption for symmetric pencils was not possible and it was left as an open question how to address this challenge. In this Part III, it is shown how the observed issue can be circumvented by using Hermitian perturbations. This leads to structure-preserving analogues of the three techniques from Parts I and II for Hermitian pencils (including real symmetric pencils) as well as for related structures. It is an important feature of these methods that the sign characteristic of the given pencil is preserved. As an application, it is shown that the resulting methods can be used to solve systems of bivariate polynomials.
Paper Structure (7 sections, 8 theorems, 69 equations, 2 tables)

This paper contains 7 sections, 8 theorems, 69 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rank-completing perturbations of Hermitian singular pencils
  4. Structure preserving projections of singular pencils
  5. Other symmetry structures
  6. Numerical experiments
  7. Conclusions

Key Result

theorem 1

Let $A,B\in\mathbb C^{n,m}$. Then there exist nonsingular matrices $P\in\mathbb C^{n,n}$ and $Q\in\mathbb C^{m,m}$ such that where $R(\lambda)=\operatorname{diag}(J-\lambda I_r,I_s-\lambda N)$ and where $J$ and $N$ are in Jordan canonical form with $N$ nilpotent and $r,s\geq 0$. Furthermore, where $L_{\eta}(\lambda)=[0 \ \, I_{\eta}]-\lambda \, [I_{\eta} \ \, 0]$ is of size $\eta\times (\eta+1)$

Theorems & Definitions (25)

  • theorem 1: Kronecker canonical form
  • theorem 2: Thompson canonical form
  • proposition 1
  • proof
  • theorem 3
  • Remark 4
  • proof
  • lemma 1
  • proof
  • theorem 5
  • ...and 15 more