In linear algebra, Weyl's inequality is a theorem about the changes to eigenvalues of a Hermitian matrix that is perturbed. It can be used to estimate the eigenvalues of a perturbed Hermitian matrix.
Weyl's inequality about perturbation Let A , B {\textstyle A,B} be Hermitian on inner product space V {\textstyle V} with dimension n {\textstyle n} , with spectrum ordered in descending order λ 1 ≥ . . . ≥ λ n {\textstyle \lambda _{1}\geq ...\geq \lambda _{n}} . Note that these eigenvalues can be ordered, because they are real (as eigenvalues of Hermitian matrices).
Weyl's inequality states that the spectrum of Hermitian matrices is stable under perturbation. Specifically, we have:
In jargon, it says that λ k {\displaystyle \lambda _{k}} is Lipschitz-continuous on the space of Hermitian matrices with operator norm.
Weyl's inequality between eigenvalues and singular values Let A ∈ C n × n {\displaystyle A\in \mathbb {C} ^{n\times n}} have singular values σ 1 ( A ) ≥ ⋯ ≥ σ n ( A ) ≥ 0 {\displaystyle \sigma _{1}(A)\geq \cdots \geq \sigma _{n}(A)\geq 0} and eigenvalues ordered so that | λ 1 ( A ) | ≥ ⋯ ≥ | λ n ( A ) | {\displaystyle |\lambda _{1}(A)|\geq \cdots \geq |\lambda _{n}(A)|} . Then
| λ 1 ( A ) ⋯ λ k ( A ) | ≤ σ 1 ( A ) ⋯ σ k ( A ) {\displaystyle |\lambda _{1}(A)\cdots \lambda _{k}(A)|\leq \sigma _{1}(A)\cdots \sigma _{k}(A)}
For k = 1 , … , n {\displaystyle k=1,\ldots ,n} , with equality for k = n {\displaystyle k=n} .
Applications
Estimating perturbations of the spectrum Let Hermitian matrices M {\displaystyle M} and N {\displaystyle N} differ by a matrix R {\displaystyle R} . Assume that R {\displaystyle R} is small in the sense that its spectral norm satisfies ‖ R ‖ 2 ≤ ϵ {\displaystyle \|R\|_{2}\leq \epsilon } for some small ϵ > 0 {\displaystyle \epsilon >0} . Then it follows that all the eigenvalues of R {\displaystyle R} are bounded in absolute value by ϵ {\displaystyle \epsilon } . Applying Weyl's inequality, it follows that the spectra of the Hermitian matrices M and N are close in the sense that
| μ i − ν i | ≤ ϵ ∀ i = 1 , … , n . {\displaystyle |\mu _{i}-\nu _{i}|\leq \epsilon \qquad \forall i=1,\ldots ,n.}
Note, however, that this eigenvalue perturbation bound is generally false for non-Hermitian matrices (or more accurately, for non-normal matrices). For a counterexample, let t > 0 {\displaystyle t>0} be arbitrarily small, and consider
M = [ 0 0 1 / t 2 0 ] , N = M + R = [ 0 1 1 / t 2 0 ] , R = [ 0 1 0 0 ] . {\displaystyle M={\begin{bmatrix}0&0\\1/t^{2}&0\end{bmatrix}},\qquad N=M+R={\begin{bmatrix}0&1\\1/t^{2}&0\end{bmatrix}},\qquad R={\begin{bmatrix}0&1\\0&0\end{bmatrix}}.}
whose eigenvalues μ 1 = μ 2 = 0 {\displaystyle \mu _{1}=\mu _{2}=0} and ν 1 = + 1 / t , ν 2 = − 1 / t {\displaystyle \nu _{1}=+1/t,\nu _{2}=-1/t} do not satisfy | μ i − ν i | ≤ ‖ R ‖ 2 = 1 {\displaystyle |\mu _{i}-\nu _{i}|\leq \|R\|_{2}=1} .
Weyl's inequality for singular values Let M {\displaystyle M} be a p × n {\displaystyle p\times n} matrix with 1 ≤ p ≤ n {\displaystyle 1\leq p\leq n} . Its singular values σ k ( M ) {\displaystyle \sigma _{k}(M)} are the p {\displaystyle p} positive eigenvalues of the ( p + n ) × ( p + n ) {\displaystyle (p+n)\times (p+n)} Hermitian augmented matrix
[ 0 M M ∗ 0 ] . {\displaystyle {\begin{bmatrix}0&M\\M^{*}&0\end{bmatrix}}.}
Therefore, Weyl's eigenvalue perturbation inequality for Hermitian matrices extends naturally to perturbation of singular values. This result gives the bound for the perturbation in the singular values of a matrix M {\displaystyle M} due to an additive perturbation Δ {\displaystyle \Delta } :
| σ k ( M + Δ ) − σ k ( M ) | ≤ σ 1 ( Δ ) {\displaystyle |\sigma _{k}(M+\Delta )-\sigma _{k}(M)|\leq \sigma _{1}(\Delta )}
where we note that the largest singular value σ 1 ( Δ ) {\displaystyle \sigma _{1}(\Delta )} coincides with the spectral norm ‖ Δ ‖ 2 {\displaystyle \|\Delta \|_{2}} .
Notes
References Matrix Theory, Joel N. Franklin, (Dover Publications, 1993) ISBN 0-486-41179-6 "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen", H. Weyl, Math. Ann., 71 (1912), 441–479