Weyl's Inequality: How Eigenvalues Move Under Perturbation

The precise statement

Let A and B be n×n Hermitian matrices (real symmetric is the special case), and order the eigenvalues of any Hermitian matrix M in descending order: λ₁(M) ≥ λ₂(M) ≥ … ≥ λₙ(M), each counted with multiplicity. Weyl's inequality states that for all indices i, j ≥ 1 with i + j − 1 ≤ n,

• λ_i+j−1(A+B) ≤ λᵢ(A) + λⱼ(B),

and, dually, whenever i + j − n ≥ 1,

• λ_i+j−n(A+B) ≥ λᵢ(A) + λⱼ(B).

The most-used consequence takes j = 1 in the upper bound and j = n in the lower bound. Since λ₁(B) ≤ ‖B‖₂ and λₙ(B) ≥ −‖B‖₂ (because ‖B‖₂ = max(|λ₁(B)|, |λₙ(B)|)), the j=1 upper bound gives λₖ(A+B) ≤ λₖ(A) + λ₁(B) ≤ λₖ(A) + ‖B‖₂ and the j=n lower bound gives λₖ(A+B) ≥ λₖ(A) + λₙ(B) ≥ λₖ(A) − ‖B‖₂, i.e. |λₖ(A+B) − λₖ(A)| ≤ ‖B‖₂ for every k. Here ‖B‖₂ is the spectral (operator 2-) norm, the largest singular value.

The picture: eigenvalues as clamped level sets

Think of each eigenvalue λₖ as a height. The Courant–Fischer min-max theorem realizes λₖ(A) as an optimization over k-dimensional subspaces: it is the best worst-case Rayleigh quotient x*Ax/(x*x) you can guarantee on some k-dimensional subspace. Adding B shifts every Rayleigh quotient by x*Bx/(x*x), a number trapped between λₙ(B) and λ₁(B).

So no matter which subspace the optimization lands on, the quotient can only wobble within a band of width ‖B‖₂ on each side. The optimization structure — a max over subspaces of a min over vectors — is monotone in the quadratic form: if you raise every value of x*Mx by at most ε, you raise λₖ by at most ε. That monotonicity is the whole story. Geometrically, the spectrum is a system of interlocked plates whose heights are all pinned to the quadratic form; you cannot lift one plate by more than you tilt the form, and B tilts it by at most ‖B‖₂.

The key idea of the proof

The engine is the Courant–Fischer min-max characterization: for Hermitian M, λₖ(M) = max_{dim V = k} min_0≠x∈V (x*Mx)/(x*x). To prove λ_i+j−1(A+B) ≤ λᵢ(A) + λⱼ(B), one uses the dual (min-max) form and a dimension-counting argument.

• Let U be the (i−1)-dimensional span of top eigenvectors of A, so on U^⊥ we have x*Ax ≤ λᵢ(A)·‖x‖². Its codimension is i−1.
• Let W be the (j−1)-dimensional top-eigenvector span of B, so on W^⊥, x*Bx ≤ λⱼ(B)·‖x‖². Codimension j−1.
• The intersection U^⊥ ∩ W^⊥ has dimension ≥ n − (i−1) − (j−1) = n − i − j + 2. On it, x*(A+B)x ≤ (λᵢ(A)+λⱼ(B))‖x‖².

Because there is a subspace of dimension n − (i+j−1) + 1 on which the Rayleigh quotient of A+B is bounded by λᵢ(A)+λⱼ(B), the min-max form forces λ_i+j−1(A+B) ≤ λᵢ(A)+λⱼ(B). The lower bound follows by applying the upper bound to −A, −B, or to A+B in place of A.

A worked example

Take A = diag(2, 0) and perturbation B = [[0, 1],[1, 0]], both Hermitian. Then λ₁(A)=2, λ₂(A)=0, and B has eigenvalues ±1, so ‖B‖₂ = 1. Weyl predicts each eigenvalue of A+B moves by at most 1.

Compute A+B = [[2, 1],[1, 0]]. Its characteristic polynomial is λ² − 2λ − 1 = 0, giving λ = 1 ± √2. So λ₁(A+B) = 1 + √2 ≈ 2.414 and λ₂(A+B) = 1 − √2 ≈ −0.414. The shifts are |2.414 − 2| ≈ 0.414 and |−0.414 − 0| ≈ 0.414 — both comfortably below the guaranteed bound of 1, and indeed equal to √2 − 1.

Check the sharp form too: with i=j=1, λ₁(A+B) ≤ λ₁(A)+λ₁(B) = 2 + 1 = 3, satisfied since 2.414 ≤ 3. Equality in |λₖ(A+B)−λₖ(A)| = ‖B‖₂ is achieved when B's extremal eigenvector aligns with the relevant eigenspace of A — e.g. A = 0, B arbitrary, where every eigenvalue moves by exactly its own value.

Why Hermitian is essential

The hypothesis that A and B are Hermitian (equivalently, real symmetric) cannot be dropped. The eigenvalues must be real and ordered for the statement to even make sense, and the min-max characterization is available only for the Hermitian case.

For non-normal matrices, eigenvalues can be wildly unstable — this is the province of pseudospectra. Consider the nilpotent Jordan block J with 1's on the superdiagonal: all n eigenvalues are 0. Perturb the bottom-left corner by ε to get J + ε·e_n e_1*. The characteristic polynomial becomes λⁿ − ε = 0, so the eigenvalues become the n-th roots ε^1/n·(roots of unity). With ε = 10⁻¹⁰ and n = 10, the eigenvalues jump to modulus 0.1 — a movement of 10⁹ times the perturbation size. Weyl's Lipschitz bound is spectacularly false here because J is not normal. This is exactly the phenomenon the Bauer–Fike theorem quantifies via the eigenvector-matrix condition number κ(V): non-orthogonal eigenvectors amplify perturbations.

Applications and significance

Weyl's inequality is a workhorse of numerical linear algebra and beyond:

• Backward stability of symmetric eigensolvers. Algorithms like the symmetric QR and divide-and-conquer compute the exact spectrum of A+E with ‖E‖₂ = O(machine ε · ‖A‖₂); Weyl then guarantees every computed eigenvalue is accurate to O(machine ε · ‖A‖₂) absolutely.
• Spectral clustering and PCA. When a data covariance or graph Laplacian is estimated with noise E, Weyl controls how far the empirical eigenvalues drift, underpinning eigenvalue-gap ('eigengap') arguments and Davis–Kahan subspace bounds.
• Random matrix theory. Weyl converts entrywise/operator-norm concentration of a noise matrix into concentration of eigenvalues, a routine step in high-dimensional statistics.
• Interlacing and inertia. Together with Cauchy interlacing it controls how spectra shift under low-rank updates, feeding Sylvester's law of inertia and eigenvalue-tracking methods.

In short, whenever you need to know that a small change to a symmetric operator produces only a small change in its energy levels, Weyl's inequality is the tool that makes it rigorous.

Weyl's inequality compared with related Hermitian eigenvalue perturbation and interlacing results

Result	Setup	Conclusion	What it controls
Weyl's inequality (full)	A, B Hermitian n×n; eigenvalues descending	λ_{i+j−1}(A+B) ≤ λᵢ(A)+λⱼ(B); also λ_{i+j−n}(A+B) ≥ λᵢ(A)+λⱼ(B)	Sums of individual eigenvalues under addition
Weyl perturbation bound (corollary)	A, E Hermitian; B=E the perturbation	|λₖ(A+E) − λₖ(A)| ≤ ‖E‖₂ for all k	Worst-case shift of a single eigenvalue
Cauchy interlacing	B = A with one row/column deleted (compression)	λₖ(A) ≥ λₖ(B) ≥ λ_{k+1}(A)	Eigenvalues of a principal submatrix
Hoffman–Wielandt	A, A+E Hermitian (or normal)	∑ₖ (λₖ(A+E)−λₖ(A))² ≤ ‖E‖_F²	Aggregate ℓ² spectral drift
Lidskii–Wielandt	A, B Hermitian	∑ₖ λ_{iₖ}(A+B) ≤ ∑ₖ λ_{iₖ}(A) + ∑ₖ λₖ(B)	Majorization of eigenvalue sums