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Analysis of eigenvalue condition numbers for a class of randomized numerical methods for singular matrix pencils

Daniel Kressner, Bor Plestenjak

TL;DR

This work addresses computing finite eigenvalues of singular matrix pencils by three randomized transformations that preserve the regular part. It develops a rigorous perturbation-based analysis using the $δ$-weak condition number and shows that the transformed pencils have eigenvalue sensitivities comparable to the original, with complex random perturbations offering superior numerical stability. The authors derive exact distributions and sharp left-tail bounds for products of random variables arising from these perturbations, and provide probabilistic guarantees that support reliable identification of well-conditioned eigenvalues. The results give both theoretical justification and practical guidance for using rank-completing modifications, normal-rank projections, and augmentation in singular generalized eigenvalue problems, including insights into when complex perturbations are advantageous.

Abstract

The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm, to that regular part. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. We show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the $δ$-weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also reconfirms that these condition numbers are a reliable criterion for detecting simple finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues. As a side result, we provide sharp left tail bounds for a product of two independent random variables distributed with the generalized beta distribution of the first kind or Kumaraswamy distribution.

Analysis of eigenvalue condition numbers for a class of randomized numerical methods for singular matrix pencils

TL;DR

This work addresses computing finite eigenvalues of singular matrix pencils by three randomized transformations that preserve the regular part. It develops a rigorous perturbation-based analysis using the -weak condition number and shows that the transformed pencils have eigenvalue sensitivities comparable to the original, with complex random perturbations offering superior numerical stability. The authors derive exact distributions and sharp left-tail bounds for products of random variables arising from these perturbations, and provide probabilistic guarantees that support reliable identification of well-conditioned eigenvalues. The results give both theoretical justification and practical guidance for using rank-completing modifications, normal-rank projections, and augmentation in singular generalized eigenvalue problems, including insights into when complex perturbations are advantageous.

Abstract

The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm, to that regular part. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. We show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the -weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also reconfirms that these condition numbers are a reliable criterion for detecting simple finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues. As a side result, we provide sharp left tail bounds for a product of two independent random variables distributed with the generalized beta distribution of the first kind or Kumaraswamy distribution.
Paper Structure (17 sections, 16 theorems, 45 equations, 1 figure, 1 table)

This paper contains 17 sections, 16 theorems, 45 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Outline
  4. Preliminaries
  5. Reducing subspaces and eigenvectors
  6. Eigenvalue perturbation theory
  7. Randomized numerical methods based on the normal rank
  8. Rank-completing modification
  9. Normal rank projections
  10. Augmentation
  11. Probabilistic analysis
  12. Preliminary results
  13. Statistics of $|\alpha|, |\beta|$
  14. Statistics of $|\alpha| \cdot |\beta|$
  15. Bounds on left tail of $|\alpha| \cdot |\beta|$
  16. ...and 2 more sections

Key Result

theorem 1

Let $A,B\in \mathbb C^{n\times n}$. Then there exist nonsingular matrices $P, Q\in \mathbb C^{n\times n}$ such that where $J$ and $N$ are in Jordan canonical form with $N$ nilpotent. Furthermore, where $L_{j}(\lambda)=[0 \ \, I_{j}]-\lambda \, [I_{j} \ \, 0]$ is of size $j\times (j+1)$, and $m_i, n_i\ge 0$ for $i=1,\dots,k$, where $k=n-\mathop{\rm nrank}\nolimits(A,B)$.

Figures (1)

  • Figure 1: Comparison of bounds and actual values of $\mathbb P(|\alpha||\beta|<t)$ for $t=10^{-5},10^{-4},10^{-3},10^{-2}$, and for $k=4$ (left) and $k=8$ (right)

Theorems & Definitions (28)

  • theorem 1: Kronecker canonical form
  • theorem 2
  • Remark 3
  • definition thmcounterdefinition
  • theorem 4
  • proposition thmcounterproposition: HMP22
  • corollary thmcountercorollary
  • proof
  • proposition thmcounterproposition: HMP22
  • corollary thmcountercorollary
  • ...and 18 more