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<head><title>eigenvalues -- find eigenvalues of a matrix</title>
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<div><h1>eigenvalues -- find eigenvalues of a matrix</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>eigenvalues M</tt></div>
</dd></dl>
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<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, or a <a href="___Mutable__Matrix.html" title="the class of all mutable matrices">MutableMatrix</a> over <a href="___R__R.html" title="the class of all real numbers">RR</a> or <a href="___C__C.html" title="the class of all complex numbers">CC</a>, which is a square n by n matrix</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Vertical__List.html">vertical list</a></span>, a list of the eigenvalues of <tt>M</tt></span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_eigenvalues_lp..._cm_sp__Hermitian_sp_eq_gt_sp..._rp.html">Hermitian => ...</a>, -- Hermitian=>true means assume the matrix is symmetric or Hermitian</span></li>
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<div class="single"><h2>Description</h2>
<div>In this example we compute the eigenvalues of a matrix and display their classes.<table class="examples"><tr><td><pre>i1 : M = matrix {{1,2}, {5,7}}
o1 = | 1 2 |
| 5 7 |
2 2
o1 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i2 : eigenvalues M
o2 = {-.358899}
{8.3589 }
o2 : VerticalList</pre>
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<tr><td><pre>i3 : class \ oo
o3 = {CC}
{CC}
o3 : VerticalList</pre>
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If the matrix is symmetric (over <a href="___R__R.html" title="the class of all real numbers">RR</a>) or Hermitian (over <a href="___C__C.html" title="the class of all complex numbers">CC</a>), this information may be provided as an optional argument <tt>Hermitian=>true</tt>, so the resulting eigenvalues will be in <a href="___R__R.html" title="the class of all real numbers">RR</a>, not <a href="___C__C.html" title="the class of all complex numbers">CC</a>.<table class="examples"><tr><td><pre>i4 : M = matrix {{1,2}, {2,1}}
o4 = | 1 2 |
| 2 1 |
2 2
o4 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i5 : eigenvalues(M, Hermitian=>true)
o5 = {-1}
{3 }
o5 : VerticalList</pre>
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<tr><td><pre>i6 : class \ oo
o6 = {RR}
{RR}
o6 : VerticalList</pre>
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The lapack library is used to compute eigenvectors of real and complex matrices.</div>
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<div class="single"><h2>Caveat</h2>
<div>The eigenvalues are approximate.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_eigenvectors.html" title="find eigenvectors of a matrix over RR or CC">eigenvectors</a> -- find eigenvectors of a matrix over RR or CC</span></li>
<li><span><a href="___S__V__D.html" title="singular value decomposition of a matrix">SVD</a> -- singular value decomposition of a matrix</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>eigenvalues</tt> :</h2>
<ul><li>eigenvalues(Matrix)</li>
<li>eigenvalues(MutableMatrix)</li>
</ul>
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