Cayley–Hamilton theorem

E621087

result in linear algebra theorem

The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.

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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 NERFINISHED ⓘ William Rowan Hamilton NERFINISHED ⓘ
proofMethods	Jordan canonical form NERFINISHED ⓘ algebraic methods ⓘ determinant-based arguments ⓘ module-theoretic arguments ⓘ
relatedTo	Jordan normal form NERFINISHED ⓘ 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 ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

linear algebra → hasKeyTheorem → Cayley–Hamilton theorem ⓘ

Arthur Cayley → notableWork → Cayley–Hamilton theorem ⓘ