Sylvester’s law of inertia
E571004
Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Sylvester’s law of inertia canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T6149922 — 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: Sylvester’s law of inertia Context triple: [James Joseph Sylvester, notableWork, Sylvester’s law of inertia]
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A.
Hermitian forms (work on quadratic forms)
Hermitian forms (work on quadratic forms) are a class of complex-valued quadratic forms that are linear in one variable and conjugate-linear in the other, generalizing real symmetric quadratic forms and playing a central role in linear algebra and functional analysis.
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B.
May–Wigner stability theorem
The May–Wigner stability theorem is a result in theoretical ecology and random matrix theory showing that large, complex systems with many random interactions are generically unstable beyond a critical level of complexity.
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C.
Hilbert’s seventeenth problem
Hilbert’s seventeenth problem is a famous question in real algebraic geometry asking whether every nonnegative polynomial can be represented as a sum of squares of rational functions.
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D.
Clebsch–Aronhold invariants
The Clebsch–Aronhold invariants are classical algebraic invariants associated with binary forms, particularly quartic forms, that play a key role in invariant theory and the classification of algebraic curves.
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E.
Cauchy interlacing 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.
- 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: Sylvester’s law of inertia Target entity description: Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
-
A.
Hermitian forms (work on quadratic forms)
Hermitian forms (work on quadratic forms) are a class of complex-valued quadratic forms that are linear in one variable and conjugate-linear in the other, generalizing real symmetric quadratic forms and playing a central role in linear algebra and functional analysis.
-
B.
May–Wigner stability theorem
The May–Wigner stability theorem is a result in theoretical ecology and random matrix theory showing that large, complex systems with many random interactions are generically unstable beyond a critical level of complexity.
-
C.
Hilbert’s seventeenth problem
Hilbert’s seventeenth problem is a famous question in real algebraic geometry asking whether every nonnegative polynomial can be represented as a sum of squares of rational functions.
-
D.
Clebsch–Aronhold invariants
The Clebsch–Aronhold invariants are classical algebraic invariants associated with binary forms, particularly quartic forms, that play a key role in invariant theory and the classification of algebraic curves.
-
E.
Cauchy interlacing 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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
theorem
ⓘ
theorem in linear algebra ⓘ |
| appliesTo |
real quadratic forms
ⓘ
real symmetric matrices ⓘ |
| assumes | the matrix is real and symmetric ⓘ |
| characterizes |
the number of negative eigenvalues of a real symmetric matrix
ⓘ
the number of positive eigenvalues of a real symmetric matrix ⓘ the number of zero eigenvalues of a real symmetric matrix ⓘ |
| concerns |
the decomposition of a real inner product space into positive, negative, and null subspaces with respect to a symmetric bilinear form
ⓘ
the multiset of eigenvalues of a real symmetric matrix counted with multiplicity ⓘ |
| equivalentTo | uniqueness of the signature of a real quadratic form up to ordering of diagonal entries ⓘ |
| field | linear algebra ⓘ |
| generalizationOf | invariance of the signature of a quadratic form ⓘ |
| holdsOver | the field of real numbers ⓘ |
| implies |
any two real symmetric matrices representing the same quadratic form in different bases have the same numbers of positive, negative, and zero eigenvalues
ⓘ
the signature of a real quadratic form is invariant under change of basis ⓘ |
| involvesConcept |
change of basis
ⓘ
congruence of matrices ⓘ definiteness of quadratic forms ⓘ eigenvalue ⓘ indefinite matrix ⓘ inertia of a matrix ⓘ negative definite matrix ⓘ nullity of a matrix ⓘ orthogonal diagonalization ⓘ positive definite matrix ⓘ rank of a matrix ⓘ real inner product space ⓘ signature of a quadratic form ⓘ symmetric bilinear form ⓘ |
| isInvariantUnder |
congruence transformation A ↦ SᵀAS with S invertible and real
ⓘ
real change of basis by invertible matrices ⓘ |
| namedAfter | James Joseph Sylvester ⓘ |
| relatedTo |
Sylvester’s criterion for positive definiteness
ⓘ
canonical form of quadratic forms ⓘ spectral theorem for real symmetric matrices ⓘ |
| statesThat |
the inertia of a real symmetric matrix is invariant under congruence transformations by invertible real matrices
ⓘ
the numbers of positive, negative, and zero eigenvalues of a real symmetric matrix are invariant under change of basis ⓘ |
| usedFor |
classifying real quadratic forms up to change of basis
ⓘ
classifying symmetric bilinear forms over the reals ⓘ determining definiteness of quadratic forms ⓘ reducing quadratic forms to canonical diagonal form ⓘ |
| usedIn |
classification of conic sections and quadrics
ⓘ
differential geometry ⓘ matrix theory ⓘ optimization theory ⓘ stability analysis in dynamical systems ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Sylvester’s law of inertia Description of subject: Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Sylvester