We know that the smith normal form of $M= \begin{pmatrix} a & b \\ c & 0 \end{pmatrix} $ is $ S=\begin{pmatrix} d & 0 \\ 0 & -bc/{d} \end{pmatrix} $ where $d=gcd(a,b,c)$. I would like to ask what the invertible matrix $U$ and $V$ that implement $S=UMV$.
For instance, we know that a diagonal matrix $M= \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} $ can written in the smith normal form $S=\begin{pmatrix} c & 0 \\ 0 & ab/c \end{pmatrix}$ by $S=\begin{pmatrix} \alpha & \beta \\ -b/c & a/c \end{pmatrix} M \begin{pmatrix} 1 & -\beta b/c \\ 1 & \alpha a/c \end{pmatrix} $ where $c=gcd(a,b)$ and $\alpha a+\beta b =c$.
Any comments will be appreciated!