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<table width="100%" summary="page for symmetric.pascal.matrix {matrixcalc}"><tr><td>symmetric.pascal.matrix {matrixcalc}</td><td style="text-align: right;">R Documentation</td></tr></table>

<h2> Symmetric Pascal matrix </h2>

<h3>Description</h3>

<p>This function returns an n by n symmetric Pascal matrix.
</p>


<h3>Usage</h3>

<pre>
symmetric.pascal.matrix(n)
</pre>


<h3>Arguments</h3>

<table summary="R argblock">
<tr valign="top"><td><code>n</code></td>
<td>
<p> Order of the matrix </p>
</td></tr>
</table>


<h3>Details</h3>

<p>In mathematics, particularly matrix theory and combinatorics, the symmetric Pascal matrix is a square matrix
from which you can derive binomial coefficients.  The matrix is an order <i>n</i> symmetric
matrix with typical element given by <i>{S_{i,j}} = {{n!} \mathord{&le;ft/
 {\vphantom {{n!} {&le;ft[ {r!\;&le;ft( {n - r} \right)!} \right]}}} \right.
 \kern-&nu;lldelimiterspace} {&le;ft[ {r!\;&le;ft( {n - r} \right)!} \right]}}</i> where
<i>n = i + j - 2</i> and <i>r = i - 1</i>.  The binomial coefficients are elegantly recovered from the
symmetric Pascal matrix by performing an <i>LU</i> decomposition as <i>{\bf{S}} = {\bf{L}}\;{\bf{U}}</i>.
</p>


<h3>Value</h3>

<p>An order <i>n</i> matrix.
</p>


<h3>Note</h3>

<p>If the argument n is not a positive integer, the function presents an error message and stops.
</p>


<h3>Author(s)</h3>

<p> Frederick Novomestky <a href="mailto:fnovomes@poly.edu">fnovomes@poly.edu</a> </p>


<h3>References</h3>

<p>Call, G. S. and D. J. Velleman, (1993). Pascal's matrices, <em>American Mathematical Monthly</em>,
April 1993, 100, 372-376.
</p>
<p>Edelman, A. and G. Strang, (2004). Pascal Matrices, <em>American Mathematical Monthly</em>,
111(3), 361-385.
</p>


<h3>Examples</h3>

<pre>
S &lt;- symmetric.pascal.matrix( 4 )
print( S )
</pre>

<hr /><div style="text-align: center;">[Package <em>matrixcalc</em> version 1.0-3 <a href="00Index.html">Index</a>]</div>
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