Weyl's inequality

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 ${\displaystyle M=N+R,\,N,}$ and ${\displaystyle R}$ be n×n Hermitian matrices, with their respective eigenvalues ${\displaystyle \mu _{i},\,\nu _{i},\,\rho _{i}}$ ordered as follows:

${\displaystyle M:\quad \mu _{1}\geq \cdots \geq \mu _{n},}$

${\displaystyle N:\quad \nu _{1}\geq \cdots \geq \nu _{n},}$

${\displaystyle R:\quad \rho _{1}\geq \cdots \geq \rho _{n}.}$

Then the following inequalities hold:

${\displaystyle \nu _{i}+\rho _{n}\leq \mu _{i}\leq \nu _{i}+\rho _{1},\quad i=1,\dots ,n,}$

and, more generally,

${\displaystyle \nu _{j}+\rho _{k}\leq \mu _{i}\leq \nu _{r}+\rho _{s},\quad j+k-n\geq i\geq r+s-1.}$

In particular, if ${\displaystyle R}$ is positive definite then plugging ${\displaystyle \rho _{n}>0}$ into the above inequalities leads to

${\displaystyle \mu _{i}>\nu _{i}\quad \forall i=1,\dots ,n.}$

Note that these eigenvalues can be ordered, because they are real (as eigenvalues of Hermitian matrices).

Weyl's inequality between eigenvalues and singular values[]

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

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

For ${\displaystyle k=1,\ldots ,n}$, with equality for ${\displaystyle k=n}$. ^[1]

Applications[]

Estimating perturbations of the spectrum[]

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

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

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 ${\displaystyle t>0}$ be arbitrarily small, and consider

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

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

Weyl's inequality for singular values[]

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

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

Therefore, Weyl's eigenvalue perturbation inequality for Hermitian matrices extends naturally to perturbation of singular values.^[3] This result gives the bound for the perturbation in the singular values of a matrix ${\displaystyle M}$ due to an additive perturbation ${\displaystyle \Delta }$:

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

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

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