The function GIsmith computes the Smith normal form S of an n by m Matrix of Gaussian integers.

If two n by n Matrices have the same Smith normal form, they are equivalent.

The Smith normal form is a diagonal Matrix where

 = number of nonzero rows (columns) of

 is in the first quadrant associate for 

 divides  for 

 divides  for all minors  of rank 

The Smith normal form is obtained by doing elementary row and column operations.  This includes interchanging rows (columns), multiplying through a row (column) by a unit in , and adding integral multiples of one row (column) to another.

In the case of three arguments, the second argument U and the third argument V will be assigned the transformation Matrices on output, such that GIsmith(A) = U . A . V.