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.
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
NERFINISHED
ⓘ
Weyl-type inequalities for normal operators NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| relatedTo |
Courant–Fischer min–max principle
NERFINISHED
ⓘ
Hoffman–Wielandt inequality NERFINISHED ⓘ Ky Fan inequalities NERFINISHED ⓘ Lidskii theorem NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.