example of Smith normal form

As an example, we will find the Smith normal form of the following matrix over the integers.

$\left(\begin{array}[]{ccc}2&4&4\\ -6&6&12\\ 10&-4&-16\end{array}\right)$

The following matrices are the intermediate steps as the algorithm is applied to the above matrix.

$\left(\begin{array}[]{ccc}2&0&0\\ -6&18&24\\ 10&-24&-36\end{array}\right)\to\left(\begin{array}[]{ccc}2&0&0\\ 0&18&24\\ 0&-24&-36\end{array}\right)$

$\to\left(\begin{array}[]{ccc}2&0&0\\ 0&18&24\\ 0&-6&-12\end{array}\right)\to\left(\begin{array}[]{ccc}2&0&0\\ 0&6&12\\ 0&18&24\end{array}\right)$

$\to\left(\begin{array}[]{ccc}2&0&0\\ 0&6&12\\ 0&0&-12\end{array}\right)\to\left(\begin{array}[]{ccc}2&0&0\\ 0&6&0\\ 0&0&12\end{array}\right)$

So the Smith normal form is

$\left(\begin{array}[]{ccc}2&0&0\\ 0&6&0\\ 0&0&12\end{array}\right)$

and the elementary divisors are $2$ , $6$ and $12$ .