Cauchy interlacing theorem

E239297

result in linear algebra 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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All labels observed (3)

Label	Occurrences
Cauchy interlacing law	1
Cauchy interlacing property	1
Cauchy interlacing theorem canonical	1

Statements (44)

Predicate	Object
instanceOf	result in linear algebra ⓘ theorem ⓘ
alsoKnownAs	Cauchy interlacing theorem ⓘ surface form: Cauchy interlacing law Cauchy interlacing theorem ⓘ surface form: Cauchy interlacing property
appearsIn	advanced linear algebra textbooks ⓘ matrix analysis literature ⓘ
appliesTo	Hermitian matrices ⓘ principal minors via their eigenvalues ⓘ symmetric matrices ⓘ
assumes	real symmetric matrix for the basic form ⓘ
category	spectral theorem consequences ⓘ
concerns	eigenvalues ⓘ principal submatrices ⓘ
field	linear algebra ⓘ matrix theory ⓘ
generalizationOf	interlacing property of roots of polynomials and their derivatives ⓘ
hasConsequence	constraints on spectra of induced subgraphs ⓘ stability of eigenvalue approximations by principal submatrices ⓘ
holdsFor	ordered eigenvalues in nonincreasing order ⓘ real eigenvalues ⓘ
implies	bounds on eigenvalues of submatrices ⓘ eigenvalues of principal submatrices lie between eigenvalues of the original matrix ⓘ monotonicity of extreme eigenvalues under taking principal submatrices ⓘ
namedAfter	Augustin-Louis Cauchy ⓘ
relatedTo	Cauchy interlacing theorem for singular values ⓘ Courant–Fischer min–max theorem ⓘ Poincaré separation theorem ⓘ Weyl inequalities ⓘ
relates	eigenvalues of a matrix ⓘ eigenvalues of its principal submatrices ⓘ
requires	ordering of eigenvalues on the real line ⓘ
typicalStatement	eigenvalues of a k×k principal submatrix interlace those of the n×n matrix with n≥k ⓘ
usedIn	control theory ⓘ dimensionality reduction ⓘ graph eigenvalue bounds ⓘ matrix perturbation theory ⓘ numerical linear algebra ⓘ PCA ⓘ surface form: principal component analysis spectral graph theory ⓘ statistics ⓘ
usedToProve	eigenvalue bounds for Laplacian matrices of graphs ⓘ interlacing families of polynomials in combinatorics ⓘ
validOver	complex numbers for Hermitian matrices ⓘ real numbers ⓘ

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Instruction

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# Requirements
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Input

Subject: Cauchy interlacing theorem
Description of subject: 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.

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Augustin-Louis Cauchy → knownFor → Cauchy interlacing theorem ⓘ

Cauchy interlacing theorem → alsoKnownAs → Cauchy interlacing theorem ⓘ

this entity surface form: Cauchy interlacing law

Cauchy interlacing theorem → alsoKnownAs → Cauchy interlacing theorem ⓘ

this entity surface form: Cauchy interlacing property