Smith Normal Form - Example

Example

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

\begin{pmatrix} 2 & 4 & 4 \\ -6 & 6 & 12 \\ 10 & -4 & -16 \end{pmatrix}

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

\to \begin{pmatrix} 2 & 0 & 0 \\ -6 & 18 & 24 \\ 10 & -24& -36 \end{pmatrix} \to \begin{pmatrix} 2 & 0 & 0 \\ 0 & 18 & 24 \\ 0 & -24& -36 \end{pmatrix}
\to \begin{pmatrix} 2 & 0 & 0 \\ 0 & 18 & 24 \\ 0 & -6 & -12 \end{pmatrix} \to \begin{pmatrix} 2 & 0 & 0 \\ 0 & 6 & 12 \\ 0 & 18 & 24 \end{pmatrix}
\to \begin{pmatrix} 2 & 0 & 0 \\ 0 & 6 & 12 \\ 0 & 0 & -12 \end{pmatrix} \to \begin{pmatrix} 2 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 12 \end{pmatrix}

So the Smith normal form is

\begin{pmatrix} 2 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 12 \end{pmatrix}

and the elementary divisors are 2, 6 and 12.

Read more about this topic:  Smith Normal Form

Famous quotes containing the word example:

    “Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
    —Marcel Proust (1871–1922)