Lambda admissible subspaces of self adjoint matrices

TL;DR

This paper introduces the concept of mbda-admissible subspaces for self-adjoint matrices, demonstrating their usefulness in low-rank approximations and eigenvalue algorithms, with theoretical bounds and numerical validation.

Contribution

It defines mbda-admissible subspaces and analyzes their properties, showing their relevance in eigenvalue approximation methods and providing bounds and numerical evidence.

Findings

mbda-admissible subspaces have favorable approximation properties.

Iterative algorithms tend to produce subspaces close to mbda-admissible subspaces.

Numerical examples confirm the advantages in clustered eigenvalue scenarios.

Abstract

Given a self-adjoint matrix $A$ and an index $h$ such that $\lambda_h(A)$ lies in a cluster of eigenvalues of $A$, we introduce the novel class of $\Lambda$-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 $\Lambda$-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 $\Lambda$-admissible subspaces. We obtain upper bounds for the distance between subspaces obtained by the Rayleigh-Ritz method applied to $A$ and the class of $\Lambda$-admissible subspaces. We also find upper bounds for the condition number of the (set-valued) map computing the class of $\Lambda$-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.