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<div class="section" id="torch-symeig">
<h1>torch.symeig<a class="headerlink" href="#torch-symeig" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="torch.symeig">
<code class="sig-prename descclassname">torch.</code><code class="sig-name descname">symeig</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">eigenvectors=False</em>, <em class="sig-param">upper=True</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.symeig" title="Permalink to this definition">¶</a></dt>
<dd><p>This function returns eigenvalues and eigenvectors
of a real symmetric matrix <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> or a batch of real symmetric matrices,
represented by a namedtuple (eigenvalues, eigenvectors).</p>
<p>This function calculates all eigenvalues (and vectors) of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>
such that <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>input</mtext><mo>=</mo><mi>V</mi><mtext>diag</mtext><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><msup><mi>V</mi><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">\text{input} = V \text{diag}(e) V^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8623000000000001em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">input</span></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:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord text"><span class="mord">diag</span></span><span class="mopen">(</span><span class="mord mathnormal">e</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span>
</span>.</p>
<p>The boolean argument <code class="xref py py-attr docutils literal notranslate"><span class="pre">eigenvectors</span></code> defines computation of
both eigenvectors and eigenvalues or eigenvalues only.</p>
<p>If it is <code class="docutils literal notranslate"><span class="pre">False</span></code>, only eigenvalues are computed. If it is <code class="docutils literal notranslate"><span class="pre">True</span></code>,
both eigenvalues and eigenvectors are computed.</p>
<p>Since the input matrix <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is supposed to be symmetric,
only the upper triangular portion is used by default.</p>
<p>If <code class="xref py py-attr docutils literal notranslate"><span class="pre">upper</span></code> is <code class="docutils literal notranslate"><span class="pre">False</span></code>, then lower triangular portion is used.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The eigenvalues are returned in ascending order. If <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is a batch of matrices,
then the eigenvalues of each matrix in the batch is returned in ascending order.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Irrespective of the original strides, the returned matrix <cite>V</cite> will
be transposed, i.e. with strides <cite>V.contiguous().transpose(-1, -2).stride()</cite>.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Extra care needs to be taken when backward through outputs. Such
operation is really only stable when all eigenvalues are distinct.
Otherwise, <code class="docutils literal notranslate"><span class="pre">NaN</span></code> can appear as the gradients are not properly defined.</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 input tensor of size <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><mo>∗</mo><mo separator="true">,</mo><mi>n</mi><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(*, n, 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">∗</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>
</span> where <cite>*</cite> is zero or more
batch dimensions consisting of symmetric matrices.</p></li>
<li><p><strong>eigenvectors</strong> (<em>boolean</em><em>, </em><em>optional</em>) – controls whether eigenvectors have to be computed</p></li>
<li><p><strong>upper</strong> (<em>boolean</em><em>, </em><em>optional</em>) – controls whether to consider upper-triangular or lower-triangular region</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 tuple of (Tensor, Tensor)</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><mo>∗</mo><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(*, m)</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">∗</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span>
</span>. The eigenvalues in ascending order.</p></li>
<li><p><strong>eigenvectors</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><mo>∗</mo><mo separator="true">,</mo><mi>m</mi><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(*, m, m)</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">∗</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span>
</span>.
If <code class="docutils literal notranslate"><span class="pre">eigenvectors=False</span></code>, it’s an empty tensor.
Otherwise, this tensor contains the orthonormal eigenvectors of the <code class="docutils literal notranslate"><span class="pre">input</span></code>.</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>Examples:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">a</span><span class="o">.</span><span class="n">t</span><span class="p">()</span> <span class="c1"># To make a symmetric</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">tensor([[-5.7827, 4.4559, -0.2344, -1.7123, -1.8330],</span>
<span class="go"> [ 4.4559, 1.4250, -2.8636, -3.2100, -0.1798],</span>
<span class="go"> [-0.2344, -2.8636, 1.7112, -5.5785, 7.1988],</span>
<span class="go"> [-1.7123, -3.2100, -5.5785, -2.6227, 3.1036],</span>
<span class="go"> [-1.8330, -0.1798, 7.1988, 3.1036, -5.1453]])</span>
<span class="gp">>>> </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">symeig</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="gp">>>> </span><span class="n">e</span>
<span class="go">tensor([-13.7012, -7.7497, -2.3163, 5.2477, 8.1050])</span>
<span class="gp">>>> </span><span class="n">v</span>
<span class="go">tensor([[ 0.1643, 0.9034, -0.0291, 0.3508, 0.1817],</span>
<span class="go"> [-0.2417, -0.3071, -0.5081, 0.6534, 0.4026],</span>
<span class="go"> [-0.5176, 0.1223, -0.0220, 0.3295, -0.7798],</span>
<span class="go"> [-0.4850, 0.2695, -0.5773, -0.5840, 0.1337],</span>
<span class="go"> [ 0.6415, -0.0447, -0.6381, -0.0193, -0.4230]])</span>
<span class="gp">>>> </span><span class="n">a_big</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a_big</span> <span class="o">=</span> <span class="n">a_big</span> <span class="o">+</span> <span class="n">a_big</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># To make a_big symmetric</span>
<span class="gp">>>> </span><span class="n">e</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">a_big</span><span class="o">.</span><span class="n">symeig</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="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">diag_embed</span><span class="p">(),</span> <span class="n">v</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">))),</span> <span class="n">a_big</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
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</dd></dl>
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