example of Smith normal form
ExampleOfSmithNormalForm
2004-05-26 12:53:29
2004-05-26 12:53:29
Example
Smith normal form
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As an example, we will find the Smith normal form of the following matrix over the integers.
\begin{equation*}
\left(\begin{array}{ccc}
2 & 4 & 4 \\
-6 & 6 & 12 \\
10 & -4 & -16
\end{array}\right)
\end{equation*}
The following matrices are the intermediate steps as the algorithm is applied to the above matrix.
\begin{equation*}
\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)
\end{equation*}
\begin{equation*}
\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)
\end{equation*}
\begin{equation*}
\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)
\end{equation*}
So the Smith normal form is
\begin{equation*}
\left(\begin{array}{ccc}
2 & 0 & 0 \\
0 & 6 & 0 \\
0 & 0 & 12
\end{array}\right)
\end{equation*}
and the elementary divisors are $2$, $6$ and $12$.