Uniform bounds for invariant subspace perturbations

Understand

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.