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

Computes the upper column Hermite normal form
together with a transformation matrix.

<p><p>

<h3>
Syntax:
</h3>
<pre>L := MatHermiteColUpperTrans(M [,"int"]);</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> MatHermiteColUpperTrans</tt> routine computes the upper
column Hermite normal form <tt> H</tt> of the integer matrix
<tt> M</tt>. Additionally it computes the rank of <tt> M</tt> and a
transformation matrix <tt> T</tt> such that <tt> H</tt> = {\tt
M}*<tt> T</tt>. The result is a list <tt> L</tt> which contains
the rank, the Hermite normal form <tt> H</tt> and the
transformation matrix <tt> T</tt>.

If the matrix is defined over a local field and has integral
coefficients, the Hermite Normal Form over the respective valuation
ring is returned.

<p><p>

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

kash> M := Mat(Z,[[3,7,10],[20,25,40]]);
[ 3  7 10]
[20 25 40]
kash> L := MatHermiteColUpperTrans(M);;
kash> L[1];
2
kash> L[2];
[0 1 0]
[0 0 5]
kash> L[3];
[ -6   1  -5]
[-16   4 -15]
[ 13  -3  12]
kash> M*L[3];
> [0 1 0]
[0 0 5]

</pre>

<p>


<hr>

<p>
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