Cauchy interlacing theorem
E239297
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Cauchy interlacing law | 1 |
| Cauchy interlacing property | 1 |
| Cauchy interlacing theorem canonical | 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 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
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: 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.
this entity surface form:
Cauchy interlacing law
this entity surface form:
Cauchy interlacing property