Weyl inequalities
E825434
Weyl inequalities are fundamental results in linear algebra that bound the eigenvalues of sums of Hermitian (or symmetric) matrices in terms of the eigenvalues of the individual matrices.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Weyl inequalities canonical | 3 |
| Weyl’s inequality | 3 |
| Weyl inequalities for singular values | 1 |
| Weyl-type inequalities for normal operators | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9844177 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Weyl inequalities Context triple: [Cauchy interlacing theorem, relatedTo, Weyl inequalities]
-
A.
Hadamard inequality
The Hadamard inequality is a fundamental result in linear algebra and analysis that bounds the absolute value of a determinant by the product of the Euclidean norms of its row or column vectors.
-
B.
Cauchy interlacing theorem
The Cauchy interlacing theorem is a fundamental result in linear algebra that relates the eigenvalues of a symmetric matrix to those of its principal submatrices, showing how they "interlace" on the real line.
-
C.
Bernstein inequalities
Bernstein inequalities are fundamental results in approximation theory and probability that provide bounds on the derivatives or deviations of functions and random variables under certain smoothness or moment conditions.
-
D.
Morawetz inequalities
Morawetz inequalities are fundamental energy and decay estimates in the study of partial differential equations, especially wave and dispersive equations, that provide control over the long-time behavior of solutions.
-
E.
Riesz rearrangement inequality
The Riesz rearrangement inequality is a fundamental result in mathematical analysis that provides an optimal bound for integrals of products of functions in terms of their symmetric decreasing rearrangements.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Weyl inequalities Target entity description: Weyl inequalities are fundamental results in linear algebra that bound the eigenvalues of sums of Hermitian (or symmetric) matrices in terms of the eigenvalues of the individual matrices.
-
A.
Hadamard inequality
The Hadamard inequality is a fundamental result in linear algebra and analysis that bounds the absolute value of a determinant by the product of the Euclidean norms of its row or column vectors.
-
B.
Cauchy interlacing theorem
The Cauchy interlacing theorem is a fundamental result in linear algebra that relates the eigenvalues of a symmetric matrix to those of its principal submatrices, showing how they "interlace" on the real line.
-
C.
Bernstein inequalities
Bernstein inequalities are fundamental results in approximation theory and probability that provide bounds on the derivatives or deviations of functions and random variables under certain smoothness or moment conditions.
-
D.
Morawetz inequalities
Morawetz inequalities are fundamental energy and decay estimates in the study of partial differential equations, especially wave and dispersive equations, that provide control over the long-time behavior of solutions.
-
E.
Riesz rearrangement inequality
The Riesz rearrangement inequality is a fundamental result in mathematical analysis that provides an optimal bound for integrals of products of functions in terms of their symmetric decreasing rearrangements.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
matrix inequality
ⓘ
result in linear algebra ⓘ theorem about eigenvalues ⓘ |
| appliesTo |
Hermitian matrices
ⓘ
symmetric matrices ⓘ |
| assumes |
A and B are n×n matrices
ⓘ
A is Hermitian ⓘ B is Hermitian ⓘ |
| category | inequalities in matrix theory ⓘ |
| concerns |
eigenvalues of matrix sums
ⓘ
spectra of Hermitian matrices ⓘ |
| field |
linear algebra
ⓘ
matrix analysis ⓘ |
| generalizationOf | eigenvalue inequalities for rank-one updates ⓘ |
| generalizes | eigenvalue interlacing inequalities ⓘ |
| gives | bounds on eigenvalues of A + B ⓘ |
| hasVariant |
Weyl inequalities for singular values
ⓘ
linked to:
Weyl inequalities
Weyl-type inequalities for normal operators ⓘ
linked to:
Weyl inequalities
|
| holdsFor |
complex Hermitian matrices
ⓘ
real symmetric matrices ⓘ |
| implies |
Lipschitz continuity of eigenvalues under perturbations
ⓘ
interlacing-type relations for eigenvalues ⓘ |λ_i(A + B) − λ_i(A)| ≤ ∥B∥_2 for each i ⓘ |
| namedAfter | Hermann Weyl ⓘ |
| relatedTo |
Courant–Fischer min–max principle
ⓘ
linked to:
Courant–Fischer min–max theorem
Hoffman–Wielandt inequality ⓘ Ky Fan inequalities ⓘ Lidskii theorem ⓘ |
| relates |
eigenvalues of A
ⓘ
eigenvalues of A + B ⓘ eigenvalues of B ⓘ |
| requires |
Hermitian matrices have real eigenvalues
ⓘ
eigenvalues are ordered nonincreasingly ⓘ |
| symbolicallyStates |
λ_i(A + B) ≤ λ_{i−k}(A) + λ_k(B) for suitable indices
ⓘ
λ_i(A + B) ≥ λ_{i−k}(A) + λ_{n−k+1}(B) for suitable indices ⓘ |
| usedIn |
control theory
ⓘ
matrix perturbation analysis ⓘ numerical linear algebra ⓘ operator theory ⓘ perturbation theory of eigenvalues ⓘ quantum mechanics ⓘ signal processing ⓘ spectral theory of Hermitian operators ⓘ |
| usedToProve |
bounds for condition numbers of eigenvalues
ⓘ
spectral inclusion results ⓘ stability results for eigenvalues ⓘ |
| uses | nonincreasing ordering of eigenvalues ⓘ |
| yearIntroducedApprox | 1910s ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Weyl inequalities Description of subject: Weyl inequalities are fundamental results in linear algebra that bound the eigenvalues of sums of Hermitian (or symmetric) matrices in terms of the eigenvalues of the individual matrices.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Weyl inequalities
linked to: Weyl inequalities
linked to: Weyl inequalities
linked to: Weyl inequalities
linked to: Weyl inequalities