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:
:$M:\quad \mu_1 \ge \cdots \ge \mu_n,$
:$N:\quad\nu_1 \ge \cdots \ge \nu_n,$
:$R:\quad\rho_1 \ge \cdots \ge \rho_n.$
Then the following inequalities hold:
:$\nu_i + \rho_n \le \mu_i \le \nu_i + \rho_1,\quad i=1,\dots,n,$
and, more generally,
:$\nu_j + \rho_k \le \mu_i \le \nu_r + \rho_s,\quad j+k-n \ge i \ge r+s-1.$
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 $A \in \mathbb^$ 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 $\, R\, _2 \le \epsilon$ 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

:$, \mu_i - \nu_i, \le \epsilon \qquad \forall i=1,\ldots,n.$

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
:$M = \begin 0 & 0 \\ 1/t^2 & 0 \end, \qquad N = M + R = \begin 0 & 1 \\ 1/t^2 & 0 \end, \qquad R = \begin 0 & 1 \\ 0 & 0 \end.$
whose eigenvalues $\mu_1 = \mu_2 = 0$ and $\nu_1 = +1/t, \nu_2 = -1/t$ do not satisfy $, \mu_i - \nu_i, \le \, R\, _2 = 1$.

Weyl's inequality for singular values

Let $M$ be a $p \times n$ matrix with $1 \le p \le n$. Its singular values $\sigma_k(M)$ are the $p$ positive eigenvalues of the $(p+n) \times (p+n)$ Hermitian augmented matrix

:$\begin 0 & M \\ M^* & 0 \end.$

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 $\Delta$:
:$, \sigma_k(M+\Delta) - \sigma_k(M), \le \sigma_1(\Delta)$
where we note that the largest singular value $\sigma_1(\Delta)$ coincides with the spectral norm $\, \Delta\, _2$.

Notes

References

* ''Matrix Theory'', Joel N. Franklin, (Dover Publications, 1993)
* "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen", H. Weyl, Math. Ann., 71 (1912), 441–479

{{DEFAULTSORT:Weyl's Inequality
Diophantine approximation
Inequalities
Linear algebra