Courant–Fischer min–max theorem

E825435

result in linear algebra result in spectral theory theorem

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.

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All labels observed (6)

Label	Occurrences
Courant–Fischer min–max theorem canonical	2
Courant–Fischer min–max principle	1
Courant–Fischer theorem	1
Courant–Fischer variational principle	1
Poincaré min–max principle	1
Rayleigh–Ritz theorem	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]

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
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 ⓘ linked to: Courant–Fischer min–max theorem Courant–Fischer variational principle ⓘ linked to: Courant–Fischer min–max theorem
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 ⓘ linked to: Rayleigh method
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 ⓘ Richard Courant ⓘ
relatedTo	Cauchy interlacing theorem ⓘ Poincaré min–max principle ⓘ linked to: Courant–Fischer min–max theorem Rayleigh–Ritz theorem ⓘ linked to: Courant–Fischer min–max theorem Weyl inequalities ⓘ
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 ⓘ min–max principle ⓘ variational characterization ⓘ

How these facts were elicited