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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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 ⓘ

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Cauchy interlacing theorem → relatedTo → Courant–Fischer min–max theorem ⓘ