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

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

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

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

Then the following inequalities hold:

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

and, more generally,

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

\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 $A\in \mathbb {C} ^{n\times n}$ 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 $k=n$ . [1]

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[2]

|\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.[3] 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 $\|\Delta \|_{2}$ .

Notes

Roger A. Horn, and Charles R. Johnson Topics in Matrix Analysis. Cambridge, 1st Edition, 1991. p.171
Weyl, Hermann. "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)." Mathematische Annalen 71, no. 4 (1912): 441-479.
Tao, Terence (2010-01-13). "254A, Notes 3a: Eigenvalues and sums of Hermitian matrices". Terence Tao's blog. Retrieved 25 May 2015.

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

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[1] Roger A. Horn, and Charles R. Johnson Topics in Matrix Analysis. Cambridge, 1st Edition, 1991. p.171

[2] Weyl, Hermann. "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)." Mathematische Annalen 71, no. 4 (1912): 441-479.

[3] Tao, Terence (2010-01-13). "254A, Notes 3a: Eigenvalues and sums of Hermitian matrices". Terence Tao's blog. Retrieved 25 May 2015.