Davis–Kahan Eigenspace Sensitivity

Concept Overview

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.

Theory

Consider a base diagonal matrix A = diag(g, -g) whereg >= 0, and an off-diagonal perturbationE = [[0, e], [e, 0]]. The perturbed matrix isA + E.

The exact eigenvalues of the perturbed matrix are±R, whereR = hypot(g, e) = √(g² + e²).

For g > 0, the upper base eigenvalue isg and its corresponding eigenspace is spanned bye_1 = [1, 0]^T. Under perturbation, the upper unit eigenvector rotates by an angleθ = arctan(e / (g + R)).

The difference in the rank-one projectors in the 2-norm is given exactly by|sin(θ)|. The Davis-Kahan bound states that|sin(θ)| <= ||E|| / δ, where δ is the separation between the base upper eigenvalue and the perturbed lower eigenvalue:δ = g - (-R) = g + R.

At g = 0, the base matrix has a repeated eigenvalue of 0. It does not identify a unique one-dimensional upper eigenspace, making the angle, projector difference, and boundUnavailable.

Worked Example

Let's consider a scenario with a Half gap of 0.5 and an Off diagonal perturbation of 0.1.

Input: g = 0.5, e = 0.1

1. Calculate Perturbed upper eigenvalue R:
   R = hypot(0.5, 0.1) = sqrt(0.5^2 + 0.1^2) = sqrt(0.25 + 0.01) = sqrt(0.26)
   R ≈ 0.5099019513592785

2. Calculate Upper eigenvalue shift:
   shift = R - g = 0.5099019513592785 - 0.5
   shift ≈ 0.009901951359278483

3. Calculate Signed eigenvector angle θ:
   θ = arctan(e / (g + R)) = arctan(0.1 / (0.5 + 0.5099019513592785))
   θ ≈ 0.09869777992494037

4. Calculate Projector difference two norm:
   norm = |sin(θ)| ≈ |sin(0.09869777992494037)|
   norm ≈ 0.09853761796664215

5. Calculate Davis-Kahan bound:
   δ = g + R = 0.5 + 0.5099019513592785 = 1.0099019513592786
   bound = |e| / δ = 0.1 / 1.0099019513592786
   bound ≈ 0.09901951359278482

Interactive Experiment

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

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.
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.
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.

Limitations

This module assumes a specific base diagonal structure and does not generalize to arbitrary matrix perturbations. It is designed to illustrate the specific Davis-Kahan bounds in two dimensions rather than provide a general eigensolver.

References

Detailed discussion on symmetric eigenspace perturbations and classical Davis-Kahan bounds can be found in the foundational literature. For a modern take and generalization, seearXiv:1405.0680.

Davis–Kahan Eigenspace Sensitivity