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Invariant subspace perturbations related to defective eigenvalues of $Δ$-Hermitian and Hamiltonian matrices

Hongguo Xu

TL;DR

This work develops structured invariant-subspace perturbation theory for Δ-Hermitian and Hamiltonian matrices when a single defective eigenvalue perturbs under a small, structure-preserving perturbation. Building on Xu's perturbation framework, it derives fractional-order eigenvalue splittings of the form $\lambda_{ij}^{(\rho)}(t)=\lambda+t^{1/\rho}\mu_{ij}^{(\rho)}+o(t^{1/\rho})$ (and their Hamiltonian analogues on the imaginary axis) and constructs corresponding invariant subspace bases that reveal how original eigenvectors and generalized eigenvectors combine in the perturbed subspaces. The analysis hinges on structured blocks and Schur-complement reductions via $S_\rho$, $\Theta_\rho(S_\rho)$, and related matrices, under the generic invertibility of $W_1,...,W_m$; a special case clarifies imaginary-axis persistence and related structure-inertia considerations. The results extend perturbation theory to defect-induced subspace changes with precise fractional-order behavior, offering insights for stability and robustness in control problems involving Hamiltonian systems. Overall, the paper provides a rigorous, symmetry-aware perturbation framework for defective eigenvalues in Δ-Hermitian and Hamiltonian matrices with practical relevance to system analysis and design.

Abstract

Structured perturbation results for invariant subspaces of $Δ$-Hermitian and Hamiltonian matrices are provided. The invariant subspaces under consideration are associated with the eigenvalues perturbed from a single defective eigenvalue. The results show how the original eigenvectors and generalized eigenvectors are involved in composing such perturbed invariant subspaces and eigenvectors.

Invariant subspace perturbations related to defective eigenvalues of $Δ$-Hermitian and Hamiltonian matrices

TL;DR

This work develops structured invariant-subspace perturbation theory for Δ-Hermitian and Hamiltonian matrices when a single defective eigenvalue perturbs under a small, structure-preserving perturbation. Building on Xu's perturbation framework, it derives fractional-order eigenvalue splittings of the form (and their Hamiltonian analogues on the imaginary axis) and constructs corresponding invariant subspace bases that reveal how original eigenvectors and generalized eigenvectors combine in the perturbed subspaces. The analysis hinges on structured blocks and Schur-complement reductions via , , and related matrices, under the generic invertibility of ; a special case clarifies imaginary-axis persistence and related structure-inertia considerations. The results extend perturbation theory to defect-induced subspace changes with precise fractional-order behavior, offering insights for stability and robustness in control problems involving Hamiltonian systems. Overall, the paper provides a rigorous, symmetry-aware perturbation framework for defective eigenvalues in Δ-Hermitian and Hamiltonian matrices with practical relevance to system analysis and design.

Abstract

Structured perturbation results for invariant subspaces of -Hermitian and Hamiltonian matrices are provided. The invariant subspaces under consideration are associated with the eigenvalues perturbed from a single defective eigenvalue. The results show how the original eigenvectors and generalized eigenvectors are involved in composing such perturbed invariant subspaces and eigenvectors.

Paper Structure

This paper contains 10 sections, 16 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Invariant subspace perturbations of $\Delta$-Hermitian matrices
  4. Nonreal eigenvalue case
  5. Real eigenvalue case
  6. Invariant subspace perturbations for Hamiltonian matrices
  7. Nonimaginary eigenvalue case
  8. Purely imaginary eigenvalue case
  9. A special case
  10. Conclusions

Key Result

Lemma 1

Consider the matrix $\lambda I_p+N+tB(t)$ with $B(t)=B+O(t)$ and $N:=N(m)$. Let $W_k$ be defined in (defw). Suppose $\rho\in\{1,\ldots,m\}$ and $W_{\rho+1}$ is invertible when $\rho<m$. Let $\gamma_1^{(\rho)},\ldots,\gamma_{s_\rho}^{(\rho)}$ be the $s_\rho$ eigenvalues of the matrix Assume $t$ is sufficiently small.

Theorems & Definitions (20)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • Proposition 3
  • Proposition 4
  • Theorem 5
  • Theorem 6
  • Corollary 7
  • Remark 2
  • Remark 3
  • ...and 10 more