Mirsky Singular-Value Stability

Worked Example

Let the base matrix A be diagonal with values 3 and 1, and apply a shear perturbation E of magnitude 1:

A = [3, 0]    E = [0, 1] [0, 1]        [0, 0] A + E = [3, 1] [0, 1]

The Gram matrix is G = (A+E)^T(A+E):

G = [3, 0] [3, 1] = [9, 3] [1, 1] [0, 1]   [3, 2]

The trace is T = 11, and the determinant is D = 9. The characteristic polynomial is λ² - 11λ + 9 = 0. We can solve for the eigenvalues (which are the squared singular values of A+E):

1. λ₁ = (11 + √(121 - 36)) / 2 ≈ 10.1097
2. λ₂ = (11 - √85) / 2 ≈ 0.8902

The singular values are σ₁ = √λ₁ ≈ 3.1795 and σ₂ = √λ₂ ≈ 0.9435. The total displacement of singular values is √((3.1795 - 3)² + (0.9435 - 1)²) ≈ 0.1882. This is strictly bounded by the Frobenius norm of E, which is 1.0.

Experiment

Use the interactive module to visualize the effects of different perturbations on the singular values:

1. Set the Direction to "Shear".
2. Set the Perturbation magnitude to "1". Observe the Singular-value displacement (0.1882) is less than the Frobenius perturbation (1.0).
3. Change the Direction to "Diagonal" with magnitude "1". The Singular-value displacement grows to 1.414, which exactly matches the Frobenius bound. This shows the bound is tight for diagonal changes.

Limitations

Mirsky's absolute stability bound holds for the singular values of arbitrary (including non-symmetric or non-square) matrices. However, it does not imply eigenvalue stability for non-normal matrices. Perturbing a highly non-normal matrix can cause massive shifts in its eigenvalues even when the singular values change by an amount bounded by the perturbation size.

References

Bernstein, D. S., Matrix Mathematics. See the section on Matrix Perturbation bounds and singular value stability assertions: https://www.stat.uchicago.edu/~lekheng/courses/309/books/Bernstein.pdf.