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<title>GNU Scientific Library – Reference Manual: Complex Hermitian Matrices</title>
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<a name="Complex-Hermitian-Matrices"></a>
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Next: <a href="Real-Nonsymmetric-Matrices.html#Real-Nonsymmetric-Matrices" accesskey="n" rel="next">Real Nonsymmetric Matrices</a>, Previous: <a href="Real-Symmetric-Matrices.html#Real-Symmetric-Matrices" accesskey="p" rel="previous">Real Symmetric Matrices</a>, Up: <a href="Eigensystems.html#Eigensystems" accesskey="u" rel="up">Eigensystems</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Complex-Hermitian-Matrices-1"></a>
<h3 class="section">15.2 Complex Hermitian Matrices</h3>
<p>For hermitian matrices, the library uses the complex form of
the symmetric bidiagonalization and QR reduction method.
</p>
<a name="index-hermitian-matrix_002c-complex_002c-eigensystem"></a>
<a name="index-complex-hermitian-matrix_002c-eigensystem"></a>
<dl>
<dt><a name="index-gsl_005feigen_005fherm_005falloc"></a>Function: <em>gsl_eigen_herm_workspace *</em> <strong>gsl_eigen_herm_alloc</strong> <em>(const size_t <var>n</var>)</em></dt>
<dd><a name="index-gsl_005feigen_005fherm_005fworkspace"></a>
<p>This function allocates a workspace for computing eigenvalues of
<var>n</var>-by-<var>n</var> complex hermitian matrices. The size of the workspace
is <em>O(3n)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005feigen_005fherm_005ffree"></a>Function: <em>void</em> <strong>gsl_eigen_herm_free</strong> <em>(gsl_eigen_herm_workspace * <var>w</var>)</em></dt>
<dd><p>This function frees the memory associated with the workspace <var>w</var>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005feigen_005fherm"></a>Function: <em>int</em> <strong>gsl_eigen_herm</strong> <em>(gsl_matrix_complex * <var>A</var>, gsl_vector * <var>eval</var>, gsl_eigen_herm_workspace * <var>w</var>)</em></dt>
<dd><p>This function computes the eigenvalues of the complex hermitian matrix
<var>A</var>. Additional workspace of the appropriate size must be provided
in <var>w</var>. The diagonal and lower triangular part of <var>A</var> are
destroyed during the computation, but the strict upper triangular part
is not referenced. The imaginary parts of the diagonal are assumed to be
zero and are not referenced. The eigenvalues are stored in the vector
<var>eval</var> and are unordered.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005feigen_005fhermv_005falloc"></a>Function: <em>gsl_eigen_hermv_workspace *</em> <strong>gsl_eigen_hermv_alloc</strong> <em>(const size_t <var>n</var>)</em></dt>
<dd><a name="index-gsl_005feigen_005fhermv_005fworkspace"></a>
<p>This function allocates a workspace for computing eigenvalues and
eigenvectors of <var>n</var>-by-<var>n</var> complex hermitian matrices. The size of
the workspace is <em>O(5n)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005feigen_005fhermv_005ffree"></a>Function: <em>void</em> <strong>gsl_eigen_hermv_free</strong> <em>(gsl_eigen_hermv_workspace * <var>w</var>)</em></dt>
<dd><p>This function frees the memory associated with the workspace <var>w</var>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005feigen_005fhermv"></a>Function: <em>int</em> <strong>gsl_eigen_hermv</strong> <em>(gsl_matrix_complex * <var>A</var>, gsl_vector * <var>eval</var>, gsl_matrix_complex * <var>evec</var>, gsl_eigen_hermv_workspace * <var>w</var>)</em></dt>
<dd><p>This function computes the eigenvalues and eigenvectors of the complex
hermitian matrix <var>A</var>. Additional workspace of the appropriate size
must be provided in <var>w</var>. The diagonal and lower triangular part of
<var>A</var> are destroyed during the computation, but the strict upper
triangular part is not referenced. The imaginary parts of the diagonal
are assumed to be zero and are not referenced. The eigenvalues are
stored in the vector <var>eval</var> and are unordered. The corresponding
complex eigenvectors are stored in the columns of the matrix <var>evec</var>.
For example, the eigenvector in the first column corresponds to the
first eigenvalue. The eigenvectors are guaranteed to be mutually
orthogonal and normalised to unit magnitude.
</p></dd></dl>
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