Smith normal form

E838595

canonical form concept in abstract algebra concept in linear algebra matrix normal form

Smith normal form is a canonical diagonal matrix form over the integers that classifies finitely generated abelian groups and simplifies solving systems of linear Diophantine equations.

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Statements (47)

Predicate	Object
instanceOf	canonical form ⓘ concept in abstract algebra ⓘ concept in linear algebra ⓘ matrix normal form ⓘ
appliesTo	integer matrices ⓘ matrices over principal ideal domains ⓘ
assumes	ring is a principal ideal domain for full theory ⓘ
characterizedBy	diagonal entries satisfying divisibility chain ⓘ each diagonal entry divides the next one ⓘ off-diagonal entries equal to zero ⓘ
definedOver	integers ⓘ principal ideal domains ⓘ
encodes	elementary divisors of a module over a PID ⓘ invariant factors of a finitely generated abelian group ⓘ
generalizes	diagonalization of integer matrices via unimodular transformations ⓘ
guarantees	existence for any integer matrix ⓘ uniqueness up to multiplication by units ⓘ
hasAlternativeName	Smith canonical form NERFINISHED ⓘ
hasDiagonalEntries	invariant factors ⓘ
hasProperty	canonical up to multiplication by units ⓘ diagonal matrix form ⓘ
implies	decomposition of finitely generated abelian groups into cyclic components ⓘ
introducedBy	Henry John Stephen Smith NERFINISHED ⓘ
involves	unimodular column operations ⓘ unimodular row operations ⓘ
obtainedBy	left and right multiplication by unimodular matrices ⓘ
relatedTo	Hermite normal form NERFINISHED ⓘ Jordan normal form NERFINISHED ⓘ elementary divisor decomposition ⓘ finitely generated modules over a PID ⓘ invariant factor decomposition ⓘ rational canonical form NERFINISHED ⓘ
usedFor	classification of finitely generated abelian groups ⓘ computing determinant up to units ⓘ computing elementary divisors ⓘ computing greatest common divisors of minors ⓘ computing invariant factors ⓘ computing invariants of modules over principal ideal domains ⓘ computing rank of an integer matrix ⓘ computing structure of abelian groups ⓘ simplifying integer linear systems ⓘ solving systems of linear Diophantine equations ⓘ
usedIn	algebraic number theory ⓘ algebraic topology ⓘ coding theory ⓘ combinatorics ⓘ computational algebra ⓘ

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

Gröbner basis → generalizationOf → Smith normal form ⓘ