Uniform bounds for invariant subspace perturbations

TL;DR

This paper introduces deterministic bounds on how invariant subspaces of symmetric matrices and their perturbations differ when measured by a row-wise metric, highlighting cases where these differences are smaller than traditional norms.

Contribution

The authors develop new bounds for invariant subspace perturbations using the two-to-infinity norm, applicable under minimal assumptions, and explore their necessity and extensions to non-normal matrices.

Findings

Row-wise perturbation bounds can be significantly smaller than traditional norms.

Necessary components of the bounds are identified, emphasizing their fundamental nature.

Extensions to non-normal matrices are briefly discussed.

Abstract

For a fixed symmetric matrix A and symmetric perturbation E we develop purely deterministic bounds on how invariant subspaces of A and A+E can differ when measured by a suitable "row-wise" metric rather than via traditional measures of subspace distance. Understanding perturbations of invariant subspaces with respect to such metrics is becoming increasingly important across a wide variety of applications and therefore necessitates new theoretical developments. Under minimal assumptions we develop new bounds on subspace perturbations under the two-to-infinity matrix norm and show in what settings these row-wise differences in the invariant subspaces can be significantly smaller than the analogous two or Frobenius norm differences. We also demonstrate that the constitutive pieces of our bounds are necessary absent additional assumptions and, therefore, our results provide a natural starting point for further analysis of specific problems. Lastly, we briefly discuss extensions of our bounds to scenarios where A and/or E are non-normal matrices.