Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.11976 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912900393730048 |
|---|---|
| author | Zuccalli, Francisco Arrieta Massey, Pedro |
| author_facet | Zuccalli, Francisco Arrieta Massey, Pedro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11976 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lambda admissible subspaces of self adjoint matrices Zuccalli, Francisco Arrieta Massey, Pedro Numerical Analysis 42C15, 15A60 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. |
| title | Lambda admissible subspaces of self adjoint matrices |
| topic | Numerical Analysis 42C15, 15A60 |
| url | https://arxiv.org/abs/2602.11976 |