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<div class="section" id="torch-eig">
<h1>torch.eig<a class="headerlink" href="#torch-eig" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="torch.eig">
<code class="sig-prename descclassname">torch.</code><code class="sig-name descname">eig</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">eigenvectors=False</em>, <em class="sig-param">*</em>, <em class="sig-param">out=None) -> (Tensor</em>, <em class="sig-param">Tensor</em><span class="sig-paren">)</span><a class="headerlink" href="#torch.eig" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the eigenvalues and eigenvectors of a real square matrix.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Since eigenvalues and eigenvectors might be complex, backward pass is supported only
for <a class="reference internal" href="torch.symeig.html#torch.symeig" title="torch.symeig"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.symeig()</span></code></a></p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>input</strong> (<a class="reference internal" href="../tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the square matrix of shape <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>×</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n \times n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>
</span> for which the eigenvalues and eigenvectors
will be computed</p></li>
<li><p><strong>eigenvectors</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.9)"><em>bool</em></a>) – <code class="docutils literal notranslate"><span class="pre">True</span></code> to compute both eigenvalues and eigenvectors;
otherwise, only eigenvalues will be computed</p></li>
</ul>
</dd>
<dt class="field-even">Keyword Arguments</dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#tuple" title="(in Python v3.9)"><em>tuple</em></a><em>, </em><em>optional</em>) – the output tensors</p>
</dd>
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><p><p>A namedtuple (eigenvalues, eigenvectors) containing</p>
<blockquote>
<div><ul class="simple">
<li><p><strong>eigenvalues</strong> (<em>Tensor</em>): Shape <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>×</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n \times 2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">)</span></span></span></span>
</span>. Each row is an eigenvalue of <code class="docutils literal notranslate"><span class="pre">input</span></code>,
where the first element is the real part and the second element is the imaginary part.
The eigenvalues are not necessarily ordered.</p></li>
<li><p><strong>eigenvectors</strong> (<em>Tensor</em>): If <code class="docutils literal notranslate"><span class="pre">eigenvectors=False</span></code>, it’s an empty tensor.
Otherwise, this tensor of shape <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>×</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n \times n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>
</span> can be used to compute normalized (unit length)
eigenvectors of corresponding eigenvalues as follows.
If the corresponding <cite>eigenvalues[j]</cite> is a real number, column <cite>eigenvectors[:, j]</cite> is the eigenvector
corresponding to <cite>eigenvalues[j]</cite>.
If the corresponding <cite>eigenvalues[j]</cite> and <cite>eigenvalues[j + 1]</cite> form a complex conjugate pair, then the
true eigenvectors can be computed as
<span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>true eigenvector</mtext><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo>=</mo><mi>e</mi><mi>i</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>v</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi><mi>s</mi><mo stretchy="false">[</mo><mo>:</mo><mo separator="true">,</mo><mi>j</mi><mo stretchy="false">]</mo><mo>+</mo><mi>i</mi><mo>×</mo><mi>e</mi><mi>i</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>v</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi><mi>s</mi><mo stretchy="false">[</mo><mo>:</mo><mo separator="true">,</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text{true eigenvector}[j] = eigenvectors[:, j] + i \times eigenvectors[:, j + 1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">true eigenvector</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord mathnormal">e</span><span class="mord mathnormal">c</span><span class="mord mathnormal">t</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mrel">:</span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.74285em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord mathnormal">e</span><span class="mord mathnormal">c</span><span class="mord mathnormal">t</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mrel">:</span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span>
</span>,
<span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>true eigenvector</mtext><mo stretchy="false">[</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo><mo>=</mo><mi>e</mi><mi>i</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>v</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi><mi>s</mi><mo stretchy="false">[</mo><mo>:</mo><mo separator="true">,</mo><mi>j</mi><mo stretchy="false">]</mo><mo>−</mo><mi>i</mi><mo>×</mo><mi>e</mi><mi>i</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>v</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi><mi>s</mi><mo stretchy="false">[</mo><mo>:</mo><mo separator="true">,</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text{true eigenvector}[j + 1] = eigenvectors[:, j] - i \times eigenvectors[:, j + 1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">true eigenvector</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord mathnormal">e</span><span class="mord mathnormal">c</span><span class="mord mathnormal">t</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mrel">:</span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.74285em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord mathnormal">e</span><span class="mord mathnormal">c</span><span class="mord mathnormal">t</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mrel">:</span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span>
</span>.</p></li>
</ul>
</div></blockquote>
</p>
</dd>
<dt class="field-even">Return type</dt>
<dd class="field-even"><p>(<a class="reference internal" href="../tensors.html#torch.Tensor" title="torch.Tensor">Tensor</a>, <a class="reference internal" href="../tensors.html#torch.Tensor" title="torch.Tensor">Tensor</a>)</p>
</dd>
</dl>
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