Smith Normal Form

The Smith normal form of an matrix over a principal ideal domain is an matrix satisfying

(1)

where and are and matrices over whose inverses also have entries in , all entries of off the main diagonal are zero, and its nonzero diagonal entries , ..., satisfy . Here means that divides in . Any remaining diagonal entries are zero, and is the rank of . The entries are the invariant factors, determined uniquely up to multiplication by units of . This form exists over every principal ideal domain, but not over every integral domain (Stanley 2016).

For an integer matrix, and are integer unimodular matrices, so their determinants are . Requiring the nonzero diagonal entries of to be positive makes the Smith normal form unique. For example,

(2)

The outer factors have determinants and 1, respectively, and , so the Smith normal form of the middle matrix has diagonal entries 2 and 4. This reduction is useful for solving linear Diophantine equations.

For the polynomial case, let be an matrix over a field . Using elementary row and column operations over the polynomial ring , the matrix (where is the identity matrix) can be put into the diagonal matrix form

(3)

where , , ..., are monic polynomials in with degrees at least one and satisfying , where means divides , which in turn divides , and so on (Dummit and Foote 2004, p. 479). The allowed operations interchange rows or columns, add a polynomial multiple of one to another, or multiply a row or column by a unit of , namely a nonzero constant in . The polynomials are the invariant factors of and determine its rational canonical form.

The Smith normal form of an integer matrix is implemented in the Wolfram Language as SmithReduce[A]. The full decomposition is given by SmithDecomposition[A], which returns satisfying . For matrices of univariate polynomials, PolynomialSmithDecomposition[M, x] gives the corresponding decomposition over a polynomial ring.