nbfigueroa/Smith-Decomposition

This toolbox computes the Smith Decomposition of any A nxm Integer and Polynomial matrix.

The Smith normal form (also called Smith Canonical form or Invariant Factor theorem) is a diagonal matrix D that contains the invariant factors of any A matrix of size n × m over a field F (in the attached implementation it is provided for the ring of integers Z and rings of polynomials F[x]).
D = |d1 0 ... 0 ... 0|= TAS
|0 d2 ... 0 ... 0|
|: : ... : ... :|
|0 0 ...dr ... 0|
|: : ... : ... :|
|0 0 ... 0 ... 0|

where d1 , ..., dr ∈ F are monic, dj |dj+1 for 1 ≤ k ≤ r − 1. T is a product of elementary row unimodular matrices, and S is a product of elementary column unimodular matrices.

Provided are two functions: [T,D,S]=smithFormInt(A) for integer matrices and [T,D,S]=smithFormPoly(A) for polynomial matrices.

Run the smithDemo.m for usage and example matrices.