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 theorem canonical	3
Cauchy interlacing law	1
Cauchy interlacing property	1

How this entity was disambiguated

This entity first appeared as the object of triple T2171659 — 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: Cauchy interlacing theorem
Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy interlacing theorem]

A. Carathéodory–Fejér interpolation
Carathéodory–Fejér interpolation is a classical result in complex analysis and approximation theory that concerns constructing analytic functions, typically with bounded or positive real part, that match prescribed initial Taylor coefficients.
B. Routh–Hurwitz stability criterion
The Routh–Hurwitz stability criterion is a mathematical test in control theory that determines whether all roots of a system’s characteristic polynomial lie in the left half of the complex plane, ensuring system stability without explicitly computing the roots.
C. Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
D. Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
E. Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
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: Cauchy interlacing theorem
Target entity description: 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.

A. Carathéodory–Fejér interpolation
Carathéodory–Fejér interpolation is a classical result in complex analysis and approximation theory that concerns constructing analytic functions, typically with bounded or positive real part, that match prescribed initial Taylor coefficients.
B. Routh–Hurwitz stability criterion
The Routh–Hurwitz stability criterion is a mathematical test in control theory that determines whether all roots of a system’s characteristic polynomial lie in the left half of the complex plane, ensuring system stability without explicitly computing the roots.
C. Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
D. Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
E. Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
F. None of above. chosen

Statements (44)

Predicate	Object
instanceOf	result in linear algebra ⓘ theorem ⓘ
alsoKnownAs	Cauchy interlacing law ⓘ linked to: Cauchy interlacing theorem Cauchy interlacing property ⓘ linked to: Cauchy interlacing theorem
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 ⓘ principal component analysis ⓘ linked to: PCA 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 ⓘ

How these facts were elicited

Referenced by (5)

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 law ⓘ

linked to: Cauchy interlacing theorem

Cauchy interlacing theorem → alsoKnownAs → Cauchy interlacing property ⓘ

linked to: Cauchy interlacing theorem

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

Poincaré separation theorem → relatedTo → Cauchy interlacing theorem ⓘ

All labels observed (3)

How this entity was disambiguated Show

Statements (44)

How these facts were elicited Show

Referenced by (5)

How this entity was disambiguated

How these facts were elicited