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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="py function">
<dt id="torch.eig">
<code class="sig-prename descclassname"><span class="pre">torch.</span></code><code class="sig-name descname"><span class="pre">eig</span></code><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eigenvectors</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></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
if eigenvalues and eigenvectors are all real valued.</p>
<p>When <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is on CUDA, <a class="reference internal" href="#torch.eig" title="torch.eig"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.eig()</span></code></a> causes
host-device synchronization.</p>
</div>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p><a class="reference internal" href="#torch.eig" title="torch.eig"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.eig()</span></code></a> is deprecated in favor of <a class="reference internal" href="torch.linalg.eig.html#torch.linalg.eig" title="torch.linalg.eig"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.eig()</span></code></a>
and will be removed in a future PyTorch release.
<a class="reference internal" href="torch.linalg.eig.html#torch.linalg.eig" title="torch.linalg.eig"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.eig()</span></code></a> returns complex tensors of dtype <cite>cfloat</cite> or <cite>cdouble</cite>
rather than real tensors mimicking complex tensors.</p>
<p><code class="docutils literal notranslate"><span class="pre">L,</span> <span class="pre">_</span> <span class="pre">=</span> <span class="pre">torch.eig(A)</span></code> should be replaced with</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">L_complex</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigvals</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
</pre></div>
</div>
<p><code class="docutils literal notranslate"><span class="pre">L,</span> <span class="pre">V</span> <span class="pre">=</span> <span class="pre">torch.eig(A,</span> <span class="pre">eigenvectors=True)</span></code> should be replaced with</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">L_complex</span><span class="p">,</span> <span class="n">V_complex</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
</pre></div>
</div>
</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:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></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 v3.10)"><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 v3.10)"><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:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></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:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></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:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></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">ec</span><span class="mord mathnormal">t</span><span class="mord mathnormal">ors</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:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7429em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></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">ec</span><span class="mord mathnormal">t</span><span class="mord mathnormal">ors</span><span class="mopen">[</span><span class="mrel">:</span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></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:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></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:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></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">ec</span><span class="mord mathnormal">t</span><span class="mord mathnormal">ors</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:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7429em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></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">ec</span><span class="mord mathnormal">t</span><span class="mord mathnormal">ors</span><span class="mopen">[</span><span class="mrel">:</span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></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>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Trivial</span> <span class="n">example</span> <span class="k">with</span> <span class="n">a</span> <span class="n">diagonal</span> <span class="n">matrix</span><span class="o">.</span> <span class="n">By</span> <span class="n">default</span><span class="p">,</span> <span class="n">only</span> <span class="n">eigenvalues</span> <span class="n">are</span> <span class="n">computed</span><span class="p">:</span>
<span class="o">>>></span> <span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">double</span><span class="p">))</span>
<span class="o">>>></span> <span class="n">e</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">e</span>
<span class="n">tensor</span><span class="p">([[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
<span class="p">[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
<span class="p">[</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">v</span>
<span class="n">tensor</span><span class="p">([],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="n">Compute</span> <span class="n">also</span> <span class="n">the</span> <span class="n">eigenvectors</span><span class="p">:</span>
<span class="o">>>></span> <span class="n">e</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">eigenvectors</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">e</span>
<span class="n">tensor</span><span class="p">([[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
<span class="p">[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
<span class="p">[</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">v</span>
<span class="n">tensor</span><span class="p">([[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
</pre></div>
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