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<a name="Tridiagonal-Decomposition-of-Real-Symmetric-Matrices"></a>
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Next: <a href="Tridiagonal-Decomposition-of-Hermitian-Matrices.html#Tridiagonal-Decomposition-of-Hermitian-Matrices" accesskey="n" rel="next">Tridiagonal Decomposition of Hermitian Matrices</a>, Previous: <a href="Modified-Cholesky-Decomposition.html#Modified-Cholesky-Decomposition" accesskey="p" rel="previous">Modified Cholesky Decomposition</a>, Up: <a href="Linear-Algebra.html#Linear-Algebra" accesskey="u" rel="up">Linear Algebra</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Tridiagonal-Decomposition-of-Real-Symmetric-Matrices-1"></a>
<h3 class="section">14.9 Tridiagonal Decomposition of Real Symmetric Matrices</h3>
<a name="index-tridiagonal-decomposition"></a>

<p>A symmetric matrix <em>A</em> can be factorized by similarity
transformations into the form,
</p>
<div class="example">
<pre class="example">A = Q T Q^T
</pre></div>

<p>where <em>Q</em> is an orthogonal matrix and <em>T</em> is a symmetric
tridiagonal matrix.
</p>
<dl>
<dt><a name="index-gsl_005flinalg_005fsymmtd_005fdecomp"></a>Function: <em>int</em> <strong>gsl_linalg_symmtd_decomp</strong> <em>(gsl_matrix * <var>A</var>, gsl_vector * <var>tau</var>)</em></dt>
<dd><p>This function factorizes the symmetric square matrix <var>A</var> into the
symmetric tridiagonal decomposition <em>Q T Q^T</em>.  On output the
diagonal and subdiagonal part of the input matrix <var>A</var> contain the
tridiagonal matrix <em>T</em>.  The remaining lower triangular part of the
input matrix contains the Householder vectors which, together with the
Householder coefficients <var>tau</var>, encode the orthogonal matrix
<em>Q</em>. This storage scheme is the same as used by <small>LAPACK</small>.  The
upper triangular part of <var>A</var> is not referenced.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005flinalg_005fsymmtd_005funpack"></a>Function: <em>int</em> <strong>gsl_linalg_symmtd_unpack</strong> <em>(const gsl_matrix * <var>A</var>, const gsl_vector * <var>tau</var>, gsl_matrix * <var>Q</var>, gsl_vector * <var>diag</var>, gsl_vector * <var>subdiag</var>)</em></dt>
<dd><p>This function unpacks the encoded symmetric tridiagonal decomposition
(<var>A</var>, <var>tau</var>) obtained from <code>gsl_linalg_symmtd_decomp</code> into
the orthogonal matrix <var>Q</var>, the vector of diagonal elements <var>diag</var>
and the vector of subdiagonal elements <var>subdiag</var>.  
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005flinalg_005fsymmtd_005funpack_005fT"></a>Function: <em>int</em> <strong>gsl_linalg_symmtd_unpack_T</strong> <em>(const gsl_matrix * <var>A</var>, gsl_vector * <var>diag</var>, gsl_vector * <var>subdiag</var>)</em></dt>
<dd><p>This function unpacks the diagonal and subdiagonal of the encoded
symmetric tridiagonal decomposition (<var>A</var>, <var>tau</var>) obtained from
<code>gsl_linalg_symmtd_decomp</code> into the vectors <var>diag</var> and <var>subdiag</var>.
</p></dd></dl>




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