Rank-One Secular Equation

Worked Example: Finding the Roots

Consider a diagonal matrix D = diag(1, 3) and a vector u = (cos θ, sin θ). We define a rank-one perturbed matrix B = D + ρ u u^T.

Let ρ = 1 and θ = 45°. Then u = (√2/2, √2/2), and u u^T = [[0.5, 0.5], [0.5, 0.5]]. Our perturbed matrix is:

B = [[1, 0], [0, 3]] + 1 * [[0.5, 0.5], [0.5, 0.5]] B = [[1.5, 0.5], [0.5, 3.5]]

To find the eigenvalues of B, we compute the characteristic polynomial:

1. Trace(B) = 1.5 + 3.5 = 5
2. Det(B) = (1.5)(3.5) - (0.5)(0.5) = 5.25 - 0.25 = 5
3. λ² - 5λ + 5 = 0

Using the quadratic formula, the eigenvalues are λ = (5 ± √(25 - 20)) / 2 = (5 ± √5) / 2.

This yields μ₁ ≈ 1.381966 and μ₂ ≈ 3.618034.

The same values are found by finding the roots of the secular equation f(λ) = 1 + ρ (u₁² / (1 - λ) + u₂² / (3 - λ)) = 0.

Experiment: Secular Poles and Updates

Open the Interactive view at https://riano.app/module/rank-one-secular-equation to observe the secular curve.

1. Set Strength rho to 0. Notice the secular poles at λ=1 and λ=3 survive as unperturbed eigenvalues.
2. Set Strength rho to 1 and Direction degrees to 45. See how the secular curve crosses zero between the poles, defining the new eigenvalues.
3. 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).
4. 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.

Limitations

The secular equation assumes non-zero vector components for all distinct eigenvalues. When u_i = 0, we must analytically remove the zero-residue term from the summation to avoid division-by-zero artifacts. This visualization only considers the 2×2 case to keep the geometry clear and bounded on screen. We also assume non-negative perturbations (ρ ≥ 0). Negative rank-one updates shift eigenvalues down instead of up.

References

Gu, M., & Eisenstat, S. C. (1994). A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem.LAPACK Working Note 132. Explains the secular equation formulation for updating eigenvalues efficiently in divide-and-conquer algorithms.