Poincaré separation theorem
E825436
The Poincaré separation theorem is a result in linear algebra and spectral theory that characterizes how the eigenvalues of a symmetric matrix relate to those of its principal submatrices, closely connected to eigenvalue interlacing phenomena.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Poincaré separation theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9844179 — 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: Poincaré separation theorem Context triple: [Cauchy interlacing theorem, relatedTo, Poincaré separation theorem]
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A.
Poincaré–Birkhoff fixed-point theorem
The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
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B.
Poincaré–Bendixson theorem
The Poincaré–Bendixson theorem is a fundamental result in the qualitative theory of dynamical systems that characterizes the possible long-term behaviors of trajectories in two-dimensional continuous flows, ruling out chaotic dynamics in the plane.
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C.
Bendixson–Dulac criterion
The Bendixson–Dulac criterion is a result in the qualitative theory of planar dynamical systems that provides conditions under which a system has no periodic orbits in a given region.
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D.
Painlevé–Kruskal theorem
The Painlevé–Kruskal theorem is a result in the theory of nonlinear differential equations that characterizes integrability through the analytic structure of their solutions, particularly via the Painlevé property.
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E.
Kolmogorov–Arnold–Moser theory
Kolmogorov–Arnold–Moser theory is a fundamental result in dynamical systems that explains the persistence of quasi-periodic motions in nearly integrable Hamiltonian systems under small perturbations.
- 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: Poincaré separation theorem Target entity description: The Poincaré separation theorem is a result in linear algebra and spectral theory that characterizes how the eigenvalues of a symmetric matrix relate to those of its principal submatrices, closely connected to eigenvalue interlacing phenomena.
-
A.
Poincaré–Birkhoff fixed-point theorem
The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
-
B.
Poincaré–Bendixson theorem
The Poincaré–Bendixson theorem is a fundamental result in the qualitative theory of dynamical systems that characterizes the possible long-term behaviors of trajectories in two-dimensional continuous flows, ruling out chaotic dynamics in the plane.
-
C.
Bendixson–Dulac criterion
The Bendixson–Dulac criterion is a result in the qualitative theory of planar dynamical systems that provides conditions under which a system has no periodic orbits in a given region.
-
D.
Painlevé–Kruskal theorem
The Painlevé–Kruskal theorem is a result in the theory of nonlinear differential equations that characterizes integrability through the analytic structure of their solutions, particularly via the Painlevé property.
-
E.
Kolmogorov–Arnold–Moser theory
Kolmogorov–Arnold–Moser theory is a fundamental result in dynamical systems that explains the persistence of quasi-periodic motions in nearly integrable Hamiltonian systems under small perturbations.
- F. None of above. chosen
Statements (40)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in linear algebra ⓘ result in spectral theory ⓘ |
| appearsIn |
texts on matrix analysis
ⓘ
texts on numerical linear algebra ⓘ |
| appliesTo |
Hermitian matrices
ⓘ
real symmetric matrices ⓘ |
| appliesWhen | considering invariant subspaces and their orthogonal complements ⓘ |
| assumes |
eigenvalues are ordered nonincreasingly or nondecreasingly
ⓘ
matrix is symmetric or Hermitian ⓘ |
| characterizes |
how eigenvalues change when restricting a symmetric operator to a subspace
ⓘ
spectral separation between a matrix and its compressions ⓘ |
| concerns |
eigenvalue interlacing
ⓘ
eigenvalues ⓘ principal submatrices ⓘ |
| describes | relationship between eigenvalues of a symmetric matrix and eigenvalues of its principal submatrices ⓘ |
| field |
linear algebra
ⓘ
matrix theory ⓘ numerical linear algebra ⓘ spectral theory ⓘ |
| generalizes | basic eigenvalue interlacing results for leading principal submatrices ⓘ |
| guarantees |
bounds on eigenvalues of principal submatrices
ⓘ
monotonicity properties of extremal eigenvalues under restriction to subspaces ⓘ |
| implies | interlacing of eigenvalues of a symmetric matrix and its principal submatrices ⓘ |
| isPartOf |
eigenvalue interlacing theory
ⓘ
theory of self-adjoint operators ⓘ |
| namedAfter | Henri Poincaré ⓘ |
| relatedTo |
Cauchy interlacing theorem
ⓘ
Courant–Fischer min–max theorem ⓘ Rayleigh–Ritz method ⓘ
linked to:
Rayleigh method
Weyl inequalities ⓘ |
| states | eigenvalues of a principal submatrix lie between eigenvalues of the original matrix in an interlacing pattern ⓘ |
| topic |
eigenvalue inequalities
ⓘ
principal minors ⓘ spectral separation ⓘ |
| usedIn |
analysis of subspace projection methods
ⓘ
approximation of eigenvalues ⓘ finite element methods ⓘ perturbation theory of eigenvalues ⓘ spectral graph theory ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Poincaré separation theorem Description of subject: The Poincaré separation theorem is a result in linear algebra and spectral theory that characterizes how the eigenvalues of a symmetric matrix relate to those of its principal submatrices, closely connected to eigenvalue interlacing phenomena.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.