Rank-One Secular Equation
Watch a positive rank-one update move eigenvalues between secular poles.
Free theory articleWatch a positive rank-one update move eigenvalues between secular poles.
Concept & interpretation
A deep dive into how a rank-one perturbation shifts eigenvalues between the poles of a secular equation.
Experiment: Secular Poles and Updates
Open the Interactive view at https://riano.app/module/rank-one-secular-equation to observe the secular curve.
- Set Strength rho to 0. Notice the secular poles at λ=1 and λ=3 survive as unperturbed eigenvalues.
- Set Strength rho to 1 and Direction degrees to 45. See how the secular curve crosses zero between the poles, defining the new eigenvalues.
- Set Direction degrees to 0. The perturbation is aligned completely with the first basis vector. The first eigenvalue moves right, while the second eigenvalue at λ=3 remains unperturbed because its corresponding term in the secular equation has a zero residue (u2=0).
- Increase Strength rho to 2 at 0 degrees. Both eigenvalues now equal 3, demonstrating that secular poles can act as strict boundaries that trap movement until terms are fully activated.