Smith normal form basis

Let $A$ be a non-zero $m\times n$ matrix over a principal ideal domain (PID). Then, there exist invertible $m\times m$ and $n\times n$ matrices $S$ and $T$, so that

$S·A·T=\left(\begin{array}{ccccccc}{\alpha }_1& 0& 0& & \cdots & & 0\\ 0& {\alpha }_2& 0& & & & 0\\ 0& 0& \ddots & & & & 0\\ ⋮& & & {\alpha }_r& & & ⋮\\ & & & & 0& & \\ & & & & & \ddots & \\ 0& & & \cdots & & & 0\end{array}\right)$

where the diagonal elements satisfy ${\alpha }_i|{\alpha }_{i+1}$. They are called invariant factors.

smith_normal_form_basis(Matrix)

Description

Given a matrix, returns a list: the first element is matrix $S$, the second element is the diagonal matrix and the third element is $T$.