Weyl Eigenvalue Perturbation Bound
Compare eigenvalue movement with a guaranteed perturbation interval.
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Weyl Eigenvalue Perturbation Bound
Compare eigenvalue movement with a guaranteed perturbation interval.
Concept Overview
Weyl's inequality provides a powerful guarantee about how much the eigenvalues of a Hermitian (or real symmetric) matrix can change when the matrix is perturbed. Specifically, if A is perturbed by a matrix E to form A + E, and we sort the eigenvalues of both matrices such that lambda_1 <= lambda_2 ... <= lambda_n, Weyl's bound states that for every index i, the shift in the eigenvalue is bounded by the spectral norm (the largest singular value) of the perturbation E: |lambda_i(A + E) - lambda_i(A)| <= ||E||_2.
Worked Example
Let's verify Weyl's inequality for a specific symmetric 2x2 matrix and perturbation.
- Define the matrices: Let the base matrix be the Coupled matrix A = [2, 1; 1, 2]. Let the perturbation be the Opposite diagonal matrix E = [1, 0; 0, -1] (so perturbation size t = 1).
- Eigenvalues of A: For A, the trace is 4 and the determinant is 2(2) - 1(1) = 3. The characteristic equation is lambda² - 4*lambda + 3 = 0. The roots are lambda_1 = 1.0 and lambda_2 = 3.0.
- Eigenvalues of A+E: The perturbed matrix is A+E = [3, 1; 1, 1]. The trace is 4 and determinant is 3(1) - 1(1) = 2. The characteristic equation is lambda² - 4*lambda + 2 = 0. Using the quadratic formula, the roots are (4 +/- sqrt(8))/2 = 2 +/- sqrt(2). So lambda~_1 ~= 0.585786 and lambda~_2 ~= 3.414214.
- Norm of E: The spectral norm of E is simply the maximum absolute value of its eigenvalues (since it's symmetric), which is ||E||_2 = 1.
- Verify the Bound: For the lower eigenvalue, the shift is |0.585786 - 1.0| = 0.414214. For the upper eigenvalue, the shift is |3.414214 - 3.0| = 0.414214. In both cases, the shift is less than or equal to the bound of 1.
Experiment
Open the Weyl Eigenvalue Perturbation Bound module to see how the eigenvalues shift under different perturbations.
- Set the Base matrix to
Repeatedto see that Weyl's bound holds even when the original matrix has repeated eigenvalues (2I). - Change the Perturbation to
Identity. Notice that adding tI shifts both eigenvalues exactly by t, making the maximum shift exactly equal to the spectral norm (bound slack is zero). - Adjust the Perturbation size using the slider to see the guaranteed bound interval (shaded region) grow and shrink, while the perturbed eigenvalues always remain within the interval.
Limitations
Weyl's bound is a global bound that depends only on the magnitude of the perturbation. It does not take into account the structure of the perturbation or the spacing between eigenvalues. In many cases, the actual shift of the eigenvalues might be much smaller than the spectral norm of E, resulting in a large "bound slack." Furthermore, while the sorted eigenvalues are guaranteed to remain within the bound, the eigenvectors may change drastically, especially when eigenvalues are closely clustered or repeated. The bound applies to additive Hermitian perturbation, which is distinct from principal-submatrix interlacing (Cauchy's interlacing theorem).
References
For more information on Weyl's inequality and other eigenvalue bounds for Hermitian matrices, see the following resource:
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Weyl Eigenvalue Perturbation Bound module.
Try Weyl Eigenvalue Perturbation Bound on Riano →