Courant–Fischer min–max theorem
E825435
The Courant–Fischer min–max theorem is a fundamental result in linear algebra and spectral theory that characterizes the eigenvalues of a Hermitian (or symmetric) matrix via variational min–max principles over subspaces.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Courant–Fischer min–max theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9844178 — 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: Courant–Fischer min–max theorem Context triple: [Cauchy interlacing theorem, relatedTo, Courant–Fischer min–max theorem]
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A.
Sylvester’s law of inertia
Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
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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.
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C.
Cayley–Hamilton theorem
The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
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D.
May–Wigner stability theorem
The May–Wigner stability theorem is a result in theoretical ecology and random matrix theory showing that large, complex systems with many random interactions are generically unstable beyond a critical level of complexity.
-
E.
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.
- 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: Courant–Fischer min–max theorem Target entity description: The Courant–Fischer min–max theorem is a fundamental result in linear algebra and spectral theory that characterizes the eigenvalues of a Hermitian (or symmetric) matrix via variational min–max principles over subspaces.
-
A.
Sylvester’s law of inertia
Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
-
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.
Cayley–Hamilton theorem
The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
-
D.
May–Wigner stability theorem
The May–Wigner stability theorem is a result in theoretical ecology and random matrix theory showing that large, complex systems with many random interactions are generically unstable beyond a critical level of complexity.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
result in linear algebra
ⓘ
result in spectral theory ⓘ theorem ⓘ |
| alsoKnownAs |
Courant–Fischer theorem
NERFINISHED
ⓘ
Courant–Fischer variational principle NERFINISHED ⓘ |
| appliesTo |
Hermitian matrices
ⓘ
complex inner product spaces ⓘ real inner product spaces ⓘ real symmetric matrices ⓘ |
| assumes |
matrix is Hermitian or real symmetric
ⓘ
matrix is diagonalizable by a unitary or orthogonal matrix ⓘ |
| characterizes |
eigenvalues of Hermitian matrices
ⓘ
eigenvalues of symmetric matrices ⓘ |
| concerns | self-adjoint linear operators in finite dimensions ⓘ |
| domain | finite-dimensional inner product spaces ⓘ |
| field |
linear algebra
ⓘ
matrix analysis ⓘ spectral theory ⓘ |
| generalizes | Rayleigh–Ritz method NERFINISHED ⓘ |
| gives |
max–min formula for k-th smallest eigenvalue
ⓘ
min–max formula for k-th largest eigenvalue ⓘ |
| hasConsequence |
eigenvalues are stationary values of Rayleigh quotient
ⓘ
extreme eigenvalues equal global extrema of Rayleigh quotient ⓘ |
| implies | ordering of eigenvalues by variational principles ⓘ |
| namedAfter |
Fritz John Fischer
NERFINISHED
ⓘ
Richard Courant NERFINISHED ⓘ |
| relatedTo |
Cauchy interlacing theorem
NERFINISHED
ⓘ
Poincaré min–max principle NERFINISHED ⓘ Rayleigh–Ritz theorem NERFINISHED ⓘ Weyl inequalities NERFINISHED ⓘ |
| relates |
eigenvalues to extremal Rayleigh quotients
ⓘ
eigenvalues to subspaces of given dimension ⓘ |
| requires | orthonormal basis of eigenvectors exists ⓘ |
| usedFor |
bounding eigenvalues of matrices
ⓘ
characterizing extremal eigenvalues ⓘ proving eigenvalue interlacing results ⓘ |
| usedIn |
eigenvalue approximation methods
ⓘ
matrix perturbation theory ⓘ numerical linear algebra ⓘ optimization over subspaces ⓘ principal component analysis ⓘ spectral analysis of graphs ⓘ spectral clustering ⓘ spectral theory of self-adjoint operators ⓘ |
| usesConcept |
Rayleigh quotient
NERFINISHED
ⓘ
min–max principle ⓘ variational characterization ⓘ |
How these facts were elicited
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Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Courant–Fischer min–max theorem Description of subject: The Courant–Fischer min–max theorem is a fundamental result in linear algebra and spectral theory that characterizes the eigenvalues of a Hermitian (or symmetric) matrix via variational min–max principles over subspaces.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.