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 $S$ where

$\mathrm{rank}\left(A\right)$ = number of nonzero rows (columns) of $S$

$S_{i,i}$ is in the first quadrant for $0<i\le \mathrm{rank}\left(A\right)$ 

$S_{i,i}$ divides $S_{i+1,i+1}$ for $0<i<\mathrm{rank}\left(A\right)$ 

$\prod _{i=1}^r{S}_{i,i}$ divides $\det \left(M\right)$ for all minors $M$ of rank  $0<r\le \mathrm{rank}\left(A\right)$ 

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 $Z_i$, 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.

$\mathrm{with}\left(\mathrm{GaussInt}\right)\&colon;$

$H≔\mathrm{Matrix}\left(\left[\left[-4+7\mathrm{I}\&comma;8+10\mathrm{I}\&comma;-6-8\mathrm{I}\right]\&comma;\left[-5+7\mathrm{I}\&comma;6-6\mathrm{I}\&comma;5\mathrm{I}\right]\&comma;\left[-10+\mathrm{I}\&comma;1-3\mathrm{I}\&comma;-10+5\mathrm{I}\right]\right]\right)$

$H≔\left[\begin{array}{ccc}\mathrm{-4}+7\mathrm{I}& 8+10\mathrm{I}& \mathrm{-6}-8\mathrm{I}\\ \mathrm{-5}+7\mathrm{I}& 6-6\mathrm{I}& 5\mathrm{I}\\ \mathrm{-10}+\mathrm{I}& 1-3\mathrm{I}& \mathrm{-10}+5\mathrm{I}\end{array}\right]$

$\mathrm{GIsmith}\left(H\right)$

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1797+791\mathrm{I}\end{array}\right]$

$A≔\mathrm{Matrix}\left(\left[\left[-4-8\mathrm{I}\&comma;-1-10\mathrm{I}\&comma;2+3\mathrm{I}\right]\&comma;\left[-1-9\mathrm{I}\&comma;8+4\mathrm{I}\&comma;-5+10\mathrm{I}\right]\right]\right)$

$A≔\left[\begin{array}{ccc}\mathrm{-4}-8\mathrm{I}& \mathrm{-1}-10\mathrm{I}& 2+3\mathrm{I}\\ \mathrm{-1}-9\mathrm{I}& 8+4\mathrm{I}& \mathrm{-5}+10\mathrm{I}\end{array}\right]$

$B≔\mathrm{GIsmith}\left(A\&comma;U\&comma;V\right)$

$B≔\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]$

$U$

$\left[\begin{array}{cc}\mathrm{-1}+4\mathrm{I}& \mathrm{-1}-\mathrm{I}\\ 5-10\mathrm{I}& 2+3\mathrm{I}\end{array}\right]$

$V$

$\left[\begin{array}{ccc}0& 43+30\mathrm{I}& 101+8\mathrm{I}\\ 0& \mathrm{-28}-29\mathrm{I}& \mathrm{-75}-21\mathrm{I}\\ 1& 66-21\mathrm{I}& 89-99\mathrm{I}\end{array}\right]$

$\mathrm{LinearAlgebra}:-\mathrm{Equal}\left(U·A·V\&comma;B\right)$

$\mathrm{true}$