Cayley–Hamilton theorem
E621087
The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cayley–Hamilton theorem canonical | 5 |
How this entity was disambiguated
This entity first appeared as the object of triple T6832964 — 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: Cayley–Hamilton theorem Context triple: [linear algebra, hasKeyTheorem, Cayley–Hamilton theorem]
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A.
Sylvester determinant
The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.
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B.
Sylvester’s law of inertia
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.
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C.
Sylvester matrix
The Sylvester matrix is a structured matrix constructed from the coefficients of two polynomials, commonly used to compute their resultant and study common roots in algebra.
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D.
Cauchy determinant
The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.
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E.
Euler’s theorem
Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
- 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: Cayley–Hamilton theorem Target entity description: The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
-
A.
Sylvester determinant
The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.
-
B.
Sylvester’s law of inertia
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.
-
C.
Sylvester matrix
The Sylvester matrix is a structured matrix constructed from the coefficients of two polynomials, commonly used to compute their resultant and study common roots in algebra.
-
D.
Cauchy determinant
The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.
-
E.
Euler’s theorem
Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
result in linear algebra
ⓘ
theorem ⓘ |
| appliesTo | square matrices ⓘ |
| category | matrix theory ⓘ |
| consequence |
every matrix satisfies a monic polynomial with coefficients in the base ring
ⓘ
matrix algebra over a field is algebraic over the base field ⓘ |
| domain | finite-dimensional vector spaces ⓘ |
| field | linear algebra ⓘ |
| generalizationOf | properties of eigenvalues of matrices ⓘ |
| historicalPeriod | 19th century ⓘ |
| holdsOver |
any commutative ring
ⓘ
fields ⓘ |
| implies |
a matrix is a root of its characteristic polynomial
ⓘ
the minimal polynomial of a matrix divides its characteristic polynomial ⓘ |
| importance | fundamental theorem in linear algebra ⓘ |
| involvesConcept |
characteristic polynomial
ⓘ
eigenvalues ⓘ endomorphisms of finite-dimensional vector spaces ⓘ linear operators ⓘ matrix algebra ⓘ minimal polynomial ⓘ |
| namedAfter |
Arthur Cayley
ⓘ
William Rowan Hamilton ⓘ |
| proofMethods |
Jordan canonical form
ⓘ
linked to:
Jordan normal form theorem
algebraic methods ⓘ determinant-based arguments ⓘ module-theoretic arguments ⓘ |
| relatedTo |
Jordan normal form
ⓘ
linked to:
Jordan normal form theorem
matrix similarity ⓘ minimal polynomial ⓘ polynomial identities in matrices ⓘ spectral theorem ⓘ |
| statement | every square matrix satisfies its own characteristic polynomial ⓘ |
| typicalFormulation | for an n×n matrix A over a commutative ring, p_A(A)=0 where p_A is the characteristic polynomial of A ⓘ |
| usedFor |
computational linear algebra algorithms
ⓘ
computing powers of matrices ⓘ control theory ⓘ differential equations involving matrices ⓘ expressing high powers of a matrix as linear combinations of lower powers ⓘ matrix function computations ⓘ proofs about eigenvalues and eigenvectors ⓘ theoretical foundation of Jordan normal form ⓘ |
| usedIn |
computations of matrix exponentials
ⓘ
engineering applications involving state-space models ⓘ systems theory ⓘ theory of linear recurrence relations ⓘ |
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: Cayley–Hamilton theorem Description of subject: The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.