Weyl's inequality

This article is about Weyl's inequality in linear algebra. For Weyl's inequality in number theory, see Weyl's inequality (number theory).

In linear algebra, Weyl's inequality is a theorem about the changes to eigenvalues of an Hermitian matrix that is perturbed. It can be used to estimate the eigenvalues of a perturbed Hermitian matrix.

Weyl's inequality about perturbation

Let ${\textstyle A}$ be Hermitian on inner product space ${\textstyle V}$ with dimension ${\textstyle n}$, with spectrum ordered in descending order ${\textstyle \lambda _{1}\geq ...\geq \lambda _{n}}$. Note that these eigenvalues can be ordered, because they are real (as eigenvalues of Hermitian matrices).

Weyl inequality — $$\lambda _{i+j-1}(A+B)\leq \lambda _{i}(A)+\lambda _{j}(B)\leq \lambda _{i+j-n}(A+B)$$

Proof

By the min-max theorem, it suffices to show that any ${\textstyle W\subset V}$ with dimension ${\textstyle i+j-1}$, there exists a unit vector ${\textstyle w}$ such that ${\textstyle \langle w,(A+B)w\rangle \leq \lambda _{i}(A)+\lambda _{j}(B)}$.

By the min-max principle, there exists some ${\textstyle W_{A}}$ with codimension ${\textstyle (i-1)}$, such that $$\lambda _{i}(A)=\max _{x\in W_{A};\|x\|=1}\langle x,Ax\rangle$$ Similarly, there exists such a ${\textstyle W_{B}}$ with codimension ${\textstyle j-1}$. Now ${\textstyle W_{A}\cap W_{B}}$ has codimension ${\textstyle \leq i+j-2}$, so it has nontrivial intersection with ${\textstyle W}$. Let ${\textstyle w\in W\cap W_{A}\cap W_{B}}$, and we have the desired vector.

The second one is a corollary of the first, by taking the negative.

Weyl's inequality states that the spectrum of Hermitian matrices is stable under perturbation. Specifically, we have:

Corollary (Spectral stability) — $$\lambda _{k}(A+B)-\lambda _{k}(A)\in [\lambda _{n}(B),\lambda _{1}(B)]$$ $$|\lambda _{k}(A+B)-\lambda _{k}(A)|\leq \|B\|_{op}$$ where
$$\|B\|_{op}=\max(|\lambda _{1}(B)|,|\lambda _{n}(B)|)$$ is the operator norm.

In jargon, it says that $\lambda _{k}$ is Lipschitz-continuous on the space of Hermitian matrices with operator norm.

Weyl's inequality between eigenvalues and singular values

Let $A\in \mathbb {C} ^{n\times n}$ have singular values $\sigma _{1}(A)\geq \cdots \geq \sigma _{n}(A)\geq 0$ and eigenvalues ordered so that $|\lambda _{1}(A)|\geq \cdots \geq |\lambda _{n}(A)|$. Then

$|\lambda _{1}(A)\cdots \lambda _{k}(A)|\leq \sigma _{1}(A)\cdots \sigma _{k}(A)$

For $k=1,\ldots ,n$, with equality for $k=n$.

Applications

Estimating perturbations of the spectrum

Assume that $R$ is small in the sense that its spectral norm satisfies $\|R\|_{2}\leq \epsilon$ for some small $\epsilon >0$. Then it follows that all the eigenvalues of $R$ are bounded in absolute value by $\epsilon$. Applying Weyl's inequality, it follows that the spectra of the Hermitian matrices M and N are close in the sense that

$|\mu _{i}-\nu _{i}|\leq \epsilon \qquad \forall i=1,\ldots ,n.$

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 $t>0$ be arbitrarily small, and consider

$M={\begin{bmatrix}0&0\\1/t^{2}&0\end{bmatrix}},\qquad N=M+R={\begin{bmatrix}0&1\\1/t^{2}&0\end{bmatrix}},\qquad R={\begin{bmatrix}0&1\\0&0\end{bmatrix}}.$

whose eigenvalues $\mu _{1}=\mu _{2}=0$ and $\nu _{1}=+1/t,\nu _{2}=-1/t$ do not satisfy $|\mu _{i}-\nu _{i}|\leq \|R\|_{2}=1$.

Weyl's inequality for singular values

Let $M$ be a $p\times n$ matrix with $1\leq p\leq n$. Its singular values $\sigma _{k}(M)$ are the $p$ positive eigenvalues of the $(p+n)\times (p+n)$ Hermitian augmented matrix

${\begin{bmatrix}0&M\\M^{*}&0\end{bmatrix}}.$

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 $M$ due to an additive perturbation $\Delta$:

$|\sigma _{k}(M+\Delta )-\sigma _{k}(M)|\leq \sigma _{1}(\Delta )$

where we note that the largest singular value $\sigma _{1}(\Delta )$ coincides with the spectral norm $\|\Delta \|_{2}$.

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