Weyl's inequality#Weyl's inequality in matrix theory

Let $A,B$ be Hermitian on inner product space $V$ with dimension $n$, with spectrum ordered in descending order $\backslash lambda\_1\; \backslash geq\; ...\; \backslash geq\; \backslash lambda\_n$. Note that these eigenvalues can be ordered, because they are real (as eigenvalues of Hermitian matrices).

{{Math theorem|name=Weyl inequality|note=|math_statement=

$$\backslash lambda\_\{i+j-1\}(A+B)\; \backslash leq\; \backslash lambda\_i(A)+\backslash lambda\_j(B)\; \backslash leq\; \backslash lambda\_\{i+j-n\}(A+B)$$

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{{Math proof|title=Proof|proof=

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

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

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

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

{{Math theorem|name=Corollary (Spectral stability)|note=|math_statement=

$$\backslash lambda\_k(A+B)\; -\; \backslash lambda\_k(A)\; \backslash in\; [\backslash lambda\_n(B),\; \backslash lambda\_1(B)]$$

$$|\backslash lambda\_k(A+B)\; -\; \backslash lambda\_k(A)|\; \backslash leq\; \backslash |B\backslash |\_\{op\}$$ where

$$\backslash |B\backslash |\_\{op\}\; =\; \backslash max(|\backslash lambda\_1(B)|,\; |\backslash lambda\_n(B)|)$$ is the operator norm.

}}

In jargon, it says that $\backslash lambda\_k$ is Lipschitz-continuous on the space of Hermitian matrices with operator norm.

Let $A\; \backslash in\; \backslash mathbb\{C\}^\{n\; \backslash times\; n\}$ have singular values $\backslash sigma\_1(A)\; \backslash geq\; \backslash cdots\; \backslash geq\; \backslash sigma\_n(A)\; \backslash geq\; 0$ and eigenvalues ordered so that $|\backslash lambda\_1(A)|\; \backslash geq\; \backslash cdots\; \backslash geq\; |\backslash lambda\_n(A)|$. Then

:$|\backslash lambda\_1(A)\; \backslash cdots\; \backslash lambda\_k(A)|\; \backslash leq\; \backslash sigma\_1(A)\; \backslash cdots\; \backslash sigma\_k(A)$

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

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

Let Hermitian matrices $M$ and $N$ differ by a matrix $R$. Assume that $R$ is small in the sense that its spectral norm satisfies $\backslash |R\backslash |\_2\; \backslash le\; \backslash epsilon$ for some small $\backslash epsilon>0$. Then it follows that all the eigenvalues of $R$ are bounded in absolute value by $\backslash epsilon$. Applying Weyl's inequality, it follows that the spectra of the Hermitian matrices M and N are close in the sense that

Weyl, Hermann. [https://link.springer.com/article/10.1007/BF01456804 "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)."] Mathematische Annalen 71, no. 4 (1912): 441-479.

:$|\backslash mu\_i\; -\; \backslash nu\_i|\; \backslash le\; \backslash epsilon\; \backslash qquad\; \backslash forall\; i=1,\backslash ldots,n.$

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

:

whose eigenvalues $\backslash mu\_1\; =\; \backslash mu\_2\; =\; 0$ and $\backslash nu\_1\; =\; +1/t,\; \backslash nu\_2\; =\; -1/t$ do not satisfy $|\backslash mu\_i\; -\; \backslash nu\_i|\; \backslash le\; \backslash |R\backslash |\_2\; =\; 1$.

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

:

Therefore, Weyl's eigenvalue perturbation inequality for Hermitian matrices extends naturally to perturbation of singular values.{{cite web|last1=Tao|first1=Terence|title=254A, Notes 3a: Eigenvalues and sums of Hermitian matrices|url=https://terrytao.wordpress.com/2010/01/12/254a-notes-3a-eigenvalues-and-sums-of-hermitian-matrices/|website=Terence Tao's blog|accessdate=25 May 2015|date=2010-01-13}} This result gives the bound for the perturbation in the singular values of a matrix $M$ due to an additive perturbation $\backslash Delta$:

:$|\backslash sigma\_k(M+\backslash Delta)\; -\; \backslash sigma\_k(M)|\; \backslash le\; \backslash sigma\_1(\backslash Delta)$

where we note that the largest singular value $\backslash sigma\_1(\backslash Delta)$ coincides with the spectral norm $\backslash |\backslash Delta\backslash |\_2$.

{{Reflist}}

• 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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Category:Diophantine approximation

Category:Inequalities (mathematics)

Category:Linear algebra