  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.
