Davis-Kahan sin θ — eigengap protects the eigenvector

unperturbed A perturbed A+E rotation wedge bound ||E||/δ
drag the orange handle on the right to steer the perturbation direction
1.50
0.50
0.20
30°
eigengap δ = λ1 − λ21.00
actual sin θ0.196
DK bound ||E|| / δ0.200
ratio (actual / bound)0.98
A is the 2×2 symmetric matrix diag(λ1, λ2); the perturbation E is a symmetric matrix of operator norm ||E|| with orientation parameter φ (specifically E = ||E||·[[cos 2φ, sin 2φ], [sin 2φ, −cos 2φ]]). The left panel plots the Rayleigh quotient R(v) = vTMv along the unit circle parametrised by angle, for M = A (grey) and M = A+E (blue); maxima are eigenvectors and the value at the maximum is the top eigenvalue. The right panel shows the top eigenvector of A (grey arrow) and of A+E (blue arrow); the red wedge is the angle between them. The Davis-Kahan bound sin θ ≤ ||E|| / δ is tight when E is in the worst-case off-diagonal direction (φ = ±45°) and otherwise slack. When ||E|| approaches δ/2, the bound saturates at 1 and the eigenvectors can swap.