I am interested to know the Smith normal form of $4 \times 2$ matrices $M$:
The two cases of my interests are:
(1). $$M_1= \begin{pmatrix} 3 & 0\\ -5 & 4\\ 4 & -5\\ 0 & 3 \end{pmatrix} $$
-
(2). $$M_2= \begin{pmatrix} 1 & 2\\ 1 & -2\\ -2 & 1\\ 2 & 1 \end{pmatrix} $$
The goal is to write: a matrix $M=\begin{pmatrix} m_{11} & m_{12}\\ m_{21} & m_{22}\\ m_{31} & m_{23}\\ m_{41} & m_{42} \end{pmatrix}$
as
$$ M=VDW=\begin{pmatrix} V_{11} & V_{12} & V_{13} & V_{14}\\ V_{21} & V_{22} & V_{23} & V_{24}\\ V_{31} & V_{23} & V_{33} & V_{34}\\ V_{41} & V_{42} & V_{43} & V_{44} \end{pmatrix} \begin{pmatrix} d_{1} & 0\\ 0 & d_{2}\\ 0 & 0\\ 0 & 0 \end{pmatrix} \begin{pmatrix} w_{11} & w_{12}\\ w_{21} & w_{22} \end{pmatrix} $$
Here $V,D,W$ are integer matrices. And $\det(V)=\det(W)=1$.
Question: what is the V,D,W matrices for $M_1$ and $M_2$?
Is there a math program one can use? thanks~ :)