Davis–Kahan Eigenspace Sensitivity

Relate eigenvector rotation to spectral separation and expose the repeated-eigenvalue boundary. When a matrix is perturbed, its eigenvectors rotate. The Davis-Kahan theorem provides a rigorous bound on this rotation angle (or the distance between corresponding spectral projectors), proportional to the size of the perturbation and inversely proportional to the spectral gap between the relevant eigenvalue clusters.

Interactive Experiment

You can observe this behavior interactively in the Davis–Kahan Eigenspace Sensitivity module.

1. Observe the default state with Half gap set to 0.5 and Off diagonal perturbation set to 0.1. Notice the small rotation angle and valid bounds.
2. Change Half gap to 0. The base eigenvalues become identical (0), making the initial eigenspace undefined. The outputs will indicate "Unavailable" as the bound is no longer meaningful.
3. Change Off diagonal perturbation to -0.1 with Half gap 0.5, and observe the rotation direction invert while the bounds and norms remain identical.