Matrix normal forms | Matrix theory

Smith normal form

In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible square matrices. In particular, the integers are a PID, so one can always calculate the Smith normal form of an integer matrix. The Smith normal form is very useful for working with finitely generated modules over a PID, and in particular for deducing the structure of a quotient of a free module. It is named after the Irish mathematician Henry John Stephen Smith. (Wikipedia).

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Frobenius normal form | Bézout's identity | Invariant factor | Topology | Up to | CW complex | Henry John Stephen Smith | Greatest common divisor | Diagonal matrix | Chain complex | Principal ideal domain | Minor (linear algebra) | Singular value decomposition | Transfer function matrix | Free module | Elementary divisors | Control theory | Homology (mathematics) | Unit (ring theory) | Hermite normal form | Bézout domain | Structure theorem for finitely generated modules over a principal ideal domain | Diophantine equation | James Whitbread Lee Glaisher | Canonical form | Unique factorization domain | Simplicial complex | Fundamental theorem of finitely generated abelian groups | Finitely generated module | Module (mathematics)