Jordan normal form theorem

E621088

mathematical theorem

The Jordan normal form theorem is a fundamental result in linear algebra that states every square matrix over an algebraically closed field is similar to a block diagonal matrix composed of Jordan blocks, providing a canonical form for linear operators.

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Observed surface forms (4)

Surface form	Occurrences
Jordan canonical form	1
Jordan decomposition	1
Jordan matrix	1
Jordan normal form	1

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ
appliesTo	linear operators on finite-dimensional vector spaces ⓘ square matrices ⓘ
assumes	algebraically closed field ⓘ
concerns	decomposition of vector spaces into generalized eigenspaces ⓘ structure of linear operators ⓘ
concludes	every linear operator on a finite-dimensional vector space over an algebraically closed field has a Jordan normal form ⓘ every square matrix is similar to a block diagonal matrix of Jordan blocks ⓘ every square matrix over an algebraically closed field is similar to a Jordan matrix ⓘ
describes	Jordan normal form NERFINISHED ⓘ
equivalentTo	classification of finite-dimensional modules over k[x] where k is algebraically closed ⓘ
field	linear algebra ⓘ matrix theory ⓘ representation theory ⓘ
hasComponentConcept	Jordan block NERFINISHED ⓘ Jordan chain ⓘ Jordan matrix NERFINISHED ⓘ
hasCondition	field must be algebraically closed for full Jordan form ⓘ
hasVariant	Jordan–Chevalley decomposition NERFINISHED ⓘ real Jordan form NERFINISHED ⓘ
historicalAttribution	Camille Jordan proved the theorem in the 19th century NERFINISHED ⓘ
implies	every linear operator decomposes into semisimple and nilpotent parts that commute ⓘ existence of a basis of generalized eigenvectors ⓘ
involves	diagonal entries equal to eigenvalues ⓘ nilpotent Jordan blocks ⓘ superdiagonal entries equal to 1 in each Jordan block ⓘ upper triangular matrices ⓘ
namedAfter	Camille Jordan NERFINISHED ⓘ
provides	canonical form for linear operators up to similarity ⓘ canonical form for matrices up to similarity ⓘ
relatesTo	characteristic polynomial ⓘ eigenvalues ⓘ eigenvectors ⓘ generalized eigenvectors ⓘ minimal polynomial ⓘ nilpotent operators ⓘ primary decomposition theorem ⓘ rational canonical form NERFINISHED ⓘ similarity of matrices ⓘ
requires	finite-dimensional vector space ⓘ
typicalField	algebraic closure of a given field ⓘ complex numbers ⓘ
usedFor	analyzing dynamical systems ⓘ classification of linear operators up to similarity ⓘ computing matrix functions ⓘ solving systems of linear differential equations ⓘ studying representations of linear transformations ⓘ

Referenced by (5)

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

linear algebra → hasKeyTheorem → Jordan normal form theorem ⓘ

Camille Jordan → knownFor → Jordan normal form theorem ⓘ

this entity surface form: Jordan normal form

Camille Jordan → knownFor → Jordan normal form theorem ⓘ

this entity surface form: Jordan decomposition

Camille Jordan → knownFor → Jordan normal form theorem ⓘ

this entity surface form: Jordan matrix

Camille Jordan → knownFor → Jordan normal form theorem ⓘ

this entity surface form: Jordan canonical form