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<head><title>eigenvectors -- find eigenvectors of a matrix over RR or CC</title>
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<div><h1>eigenvectors -- find eigenvectors of a matrix over RR or CC</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>(eigvals, eigvecs) = eigenvectors M</tt></div>
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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><tt>eigvals</tt>, <span>a <a href="___Vertical__List.html">vertical list</a></span>, a list of the eigenvalues of <tt>M</tt></span></li>
<li><span><tt>eigvecs</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, or <span>a <a href="___Mutable__Matrix.html">mutable matrix</a></span>, if <tt>M</tt> is one), whose columns are the corresponding eigenvectors 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="_eigenvectors_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>The resulting matrix is over <a href="___C__C.html" title="the class of all complex numbers">CC</a>, and contains the eigenvectors of <tt>M</tt>. The lapack library is used to compute eigenvectors of real and complex matrices.<p/>
Recall that if <tt>v</tt> is a non-zero vector such that <tt>Mv = av</tt>, for a scalar a, then <tt>v</tt> is called an eigenvector corresponding to the eigenvalue <tt>a</tt>.<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 : eigenvectors M
o2 = ({-.358899}, | -.827138 -.262266 |)
{8.3589 } | .561999 -.964996 |
o2 : Sequence</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 should be provided as an optional argument <tt>Hermitian=>true</tt>. In this case, the resulting eigenvalues will be returned as real numbers, and if <tt>M</tt> is real, the matrix of of eigenvectors will be real.<table class="examples"><tr><td><pre>i3 : M = matrix {{1, 2}, {2, 1}}
o3 = | 1 2 |
| 2 1 |
2 2
o3 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i4 : (e,v) = eigenvectors(M, Hermitian=>true)
o4 = ({-1}, | -.707107 .707107 |)
{3 } | .707107 .707107 |
o4 : Sequence</pre>
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<tr><td><pre>i5 : class \ e
o5 = {RR}
{RR}
o5 : VerticalList</pre>
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<tr><td><pre>i6 : v
o6 = | -.707107 .707107 |
| .707107 .707107 |
2 2
o6 : Matrix RR <--- RR
53 53</pre>
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<div class="single"><h2>Caveat</h2>
<div>The eigenvectors are approximate.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_eigenvalues.html" title="find eigenvalues of a matrix">eigenvalues</a> -- find eigenvalues of a matrix</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>
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<div class="waystouse"><h2>Ways to use <tt>eigenvectors</tt> :</h2>
<ul><li>eigenvectors(Matrix)</li>
<li>eigenvectors(MutableMatrix)</li>
</ul>
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