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<p>&nbsp;<p>
<hr noshade size="5">
<h2><a name="MatSmithTrans"></a>
MatSmithTrans
</h2>

Computes the Smith normal form together with transformation matrices.

<p><p>

<h3>
Syntax:
</h3>
<pre>L := MatSmithTrans(M);</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td></td>
</tr>
<tr>
<td>matrix</td>
<td><pre>  M  </pre></td>
<td></td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
The <tt> MatSmithTrans</tt> routine computes the Smith normal form
<tt> S</tt> of the integer matrix <tt> M</tt>. Additionally it
computes the rank of <tt> M</tt> and transformation matrices
<tt> A</tt> and <tt> B</tt> such that <tt> S</tt> = <tt> A</tt>*{\tt
M}*<tt> B</tt>. The result is a list <tt> L</tt> which contains
the rank, the Smith normal form <tt> S</tt> and the
transformation matrices <tt> A</tt> and <tt> B</tt>.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Compute the upper column Smith normal form of
\left(\begin{array}{ccc} 3 & 7 & 10 20 & 25 & 40
 17 & 6 & 9 \end{array}right). 
<pre>

kash> M := Mat(Z,[[3,7,10],[20,25,40],[17,6,9]]);
[ 3  7 10]
[20 25 40]
[17  6  9]
kash> L := MatSmithTrans(M);;
kash> rank := L[1];
3
kash> S := L[2];
[  1   0   0]
[  0   1   0]
[  0   0 405]
kash> A := L[3];
[  1   0   0]
[-56 -11   4]
[-70 -14   5]
kash> B := L[4];
[   1    1 -318]
[   4    1 -308]
[  -3   -1  311]
kash> A*M*B;
> [  1   0   0]
[  0   1   0]
[  0   0 405]

</pre>

<p>


<hr>

<p>
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