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Lambda admissible subspaces of self adjoint matrices

Francisco Arrieta Zuccalli, Pedro Massey

TL;DR

The paper develops a robust framework for analyzing low-rank, self-adjoint approximations in the presence of eigenvalue clusters by introducing $\Lambda$-admissible subspaces. It shows that compressions $P_{\mathcal S}AP_{\mathcal S}$ onto these subspaces retain near-optimal spectral properties and connects these subspaces to standard iterative methods (SIM, Krylov) and the Rayleigh-Ritz procedure. The authors derive explicit bounds on distances to the $\Lambda$-admissible class, provide computable approximations, quantify Ritz-related proximity, and establish a condition-number-style sensitivity measure for the subspace class, all with supporting numerical experiments. The results suggest that focusing on the $\Lambda$-admissible class improves stability and accuracy in clustered eigenvalue settings, offering new tools for efficient, reliable eigenvalue computations in large-scale Hermitian problems.

Abstract

Given a self-adjoint matrix $A$ and an index $h$ such that $λ_h(A)$ lies in a cluster of eigenvalues of $A$, we introduce the novel class of $Λ$-admissible subspaces of $A$ of dimension $h$. First, we show that the low-rank approximation of the form $P_{\mathcal{T}} A P_{\mathcal{T}}$, for a subspace $\mathcal{T}$ that is close to any $Λ$-admissible subspace of $A$, has nice properties. Then, we prove that some well-known iterative algorithms (such as the Subspace Iteration Method, or the Krylov subspace method) produce subspaces that become arbitrarily close to $Λ$-admissible subspaces. We obtain upper bounds for the distance between subspaces obtained by the Rayleigh-Ritz method applied to $A$ and the class of $Λ$-admissible subspaces. We also find upper bounds for the condition number of the (set-valued) map computing the class of $Λ$-admissible subspaces of $A$. Finally, we include numerical examples that show the advantage of considering this new class of subspaces in the clustered eigenvalue setting.

Lambda admissible subspaces of self adjoint matrices

TL;DR

The paper develops a robust framework for analyzing low-rank, self-adjoint approximations in the presence of eigenvalue clusters by introducing -admissible subspaces. It shows that compressions onto these subspaces retain near-optimal spectral properties and connects these subspaces to standard iterative methods (SIM, Krylov) and the Rayleigh-Ritz procedure. The authors derive explicit bounds on distances to the -admissible class, provide computable approximations, quantify Ritz-related proximity, and establish a condition-number-style sensitivity measure for the subspace class, all with supporting numerical experiments. The results suggest that focusing on the -admissible class improves stability and accuracy in clustered eigenvalue settings, offering new tools for efficient, reliable eigenvalue computations in large-scale Hermitian problems.

Abstract

Given a self-adjoint matrix and an index such that lies in a cluster of eigenvalues of , we introduce the novel class of -admissible subspaces of of dimension . First, we show that the low-rank approximation of the form , for a subspace that is close to any -admissible subspace of , has nice properties. Then, we prove that some well-known iterative algorithms (such as the Subspace Iteration Method, or the Krylov subspace method) produce subspaces that become arbitrarily close to -admissible subspaces. We obtain upper bounds for the distance between subspaces obtained by the Rayleigh-Ritz method applied to and the class of -admissible subspaces. We also find upper bounds for the condition number of the (set-valued) map computing the class of -admissible subspaces of . Finally, we include numerical examples that show the advantage of considering this new class of subspaces in the clustered eigenvalue setting.
Paper Structure (17 sections, 14 theorems, 125 equations, 5 figures, 1 algorithm)

This paper contains 17 sections, 14 theorems, 125 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Principal angles between subspaces
  5. Dominant eigenspaces of Hermitian matrices
  6. Introducing Λ-admissible subspaces
  7. Main results
  8. A bound for the distance to the class of Λ-admissible subspaces
  9. Computable approximations of Λ-admissible subspaces
  10. Rayleigh-Ritz method and Λ-admissible subspaces
  11. On the sensitivity of Λ-admissible subspaces computation
  12. Numerical examples
  13. Several proofs
  14. Proof of Theorem \ref{['teo dist a adm2']}
  15. Proof of Theorem \ref{['teo se aprox a admis']}
  16. ...and 2 more sections

Key Result

Theorem 4.2

Consider Notation nota sec 4. Let ${\cal T}\subset\mathbb{K}^n$ be such that $\Theta(\mathcal{X}_k,{\cal T})<\frac{\pi}{2}\, I_{ \min\{t,k\}}$. Then, for every unitarily invariant norm ${\left\| \cdot \right\|}$ we have that

Figures (5)

  • Figure 1: Exponential decay model with $\delta\approx10^{-3}$ and $\gamma\approx1$
  • Figure 2: Exponential decay model with $\delta\approx10^{-4}$ and $\gamma\approx1$
  • Figure 3: Exponential decay model with $\delta\approx10^{-3}$ and $\gamma\approx0.4$
  • Figure 4: Exponential decay model with $\delta\approx10^{-3}$ and $\gamma\approx1$
  • Figure 5: Linear decay model with $\delta\approx10^{-3}$ and $\gamma\approx1$

Theorems & Definitions (53)

  • Remark 2.1
  • Definition 2.3
  • Remark 2.4: Dominant eigenspaces and eigendecompositions
  • Remark 2.5: Dominant eigenspaces and eigenvalue approximations
  • Remark 2.6: Dominant eigenspaces and low-rank approximations
  • Definition 3.1
  • Remark 3.2: When to - and why - consider $\Lambda$-admissible subspaces
  • Remark 3.3: $\Lambda$-admissible subspaces and eigenvalue approximation
  • Remark 3.4: $\Lambda$-admissible subspaces and low-rank approximations
  • Remark 3.5: Stability of $\Lambda$-admissible subspaces under the action of $A$
  • ...and 43 more