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<li class="toctree-l1"><a class="reference internal" href="notes/amp_examples.html">Automatic Mixed Precision examples</a></li>
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<div class="section" id="torch-linalg">
<h1>torch.linalg<a class="headerlink" href="#torch-linalg" title="Permalink to this headline">¶</a></h1>
<p>Common linear algebra operations.</p>
<p>This module is in BETA. New functions are still being added, and some
functions may change in future PyTorch releases. See the documentation of each
function for details.</p>
<span class="target" id="module-torch.linalg"></span><div class="section" id="functions">
<h2>Functions<a class="headerlink" href="#functions" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="torch.linalg.cholesky">
<code class="sig-prename descclassname">torch.linalg.</code><code class="sig-name descname">cholesky</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">*</em>, <em class="sig-param">out=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.linalg.cholesky" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the Cholesky decomposition of a Hermitian (or symmetric for real-valued matrices)
positive-definite matrix or the Cholesky decompositions for a batch of such matrices.
Each decomposition has the form:</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>input</mtext><mo>=</mo><mi>L</mi><msup><mi>L</mi><mi>H</mi></msup></mrow><annotation encoding="application/x-tex">\text{input} = LL^H</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:0.8913309999999999em;vertical-align:0em;"></span><span class="mord mathnormal">L</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;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:0.08125em;">H</span></span></span></span></span></span></span></span></span></span></span></span>
</div><p>where <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">L</span></span></span></span>
</span> is a lower-triangular matrix and <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>L</mi><mi>H</mi></msup></mrow><annotation encoding="application/x-tex">L^H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413309999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal">L</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:0.08125em;">H</span></span></span></span></span></span></span></span></span></span></span>
</span> is the conjugate transpose of <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">L</span></span></span></span>
</span>,
which is just a transpose for the case of real-valued input matrices.
In code it translates to <code class="docutils literal notranslate"><span class="pre">input</span> <span class="pre">=</span> <span class="pre">L</span> <span class="pre">@</span> <span class="pre">L.t()</span></code> if <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is real-valued and
<code class="docutils literal notranslate"><span class="pre">input</span> <span class="pre">=</span> <span class="pre">L</span> <span class="pre">@</span> <span class="pre">L.conj().t()</span></code> if <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is complex-valued.
The batch of <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">L</span></span></span></span>
</span> matrices is returned.</p>
<p>Supports real-valued and complex-valued inputs.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>When given inputs on a CUDA device, this function synchronizes that device with the CPU.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>LAPACK’s <cite>potrf</cite> is used for CPU inputs, and MAGMA’s <cite>potrf</cite> is used for CUDA inputs.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>If <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is not a Hermitian positive-definite matrix, or if it’s a batch of matrices
and one or more of them is not a Hermitian positive-definite matrix, then a RuntimeError will be thrown.
If <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is a batch of matrices, then the error message will include the batch index
of the first matrix that is not Hermitian positive-definite.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><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> consisting of Hermitian positive-definite
<span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></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:0.66666em;vertical-align:-0.08333em;"></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:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span></span></span></span>
</span> matrices, where <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.46528em;vertical-align:0em;"></span><span class="mord">∗</span></span></span></span>
</span> is zero or more batch dimensions.</p>
</dd>
<dt class="field-even">Keyword Arguments</dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – The output tensor. Ignored if <code class="docutils literal notranslate"><span class="pre">None</span></code>. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></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">2</span><span class="p">,</span> <span class="mi">2</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">complex128</span><span class="p">)</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">mm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">t</span><span class="p">()</span><span class="o">.</span><span class="n">conj</span><span class="p">())</span> <span class="c1"># creates a Hermitian positive-definite matrix</span>
<span class="gp">>>> </span><span class="n">l</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">cholesky</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">tensor([[2.5266+0.0000j, 1.9586-2.0626j],</span>
<span class="go"> [1.9586+2.0626j, 9.4160+0.0000j]], dtype=torch.complex128)</span>
<span class="gp">>>> </span><span class="n">l</span>
<span class="go">tensor([[1.5895+0.0000j, 0.0000+0.0000j],</span>
<span class="go"> [1.2322+1.2976j, 2.4928+0.0000j]], dtype=torch.complex128)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">mm</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">l</span><span class="o">.</span><span class="n">t</span><span class="p">()</span><span class="o">.</span><span class="n">conj</span><span class="p">())</span>
<span class="go">tensor([[2.5266+0.0000j, 1.9586-2.0626j],</span>
<span class="go"> [1.9586+2.0626j, 9.4160+0.0000j]], dtype=torch.complex128)</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">3</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="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="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">a</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"># creates a symmetric positive-definite matrix</span>
<span class="gp">>>> </span><span class="n">l</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">cholesky</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">tensor([[[ 1.1629, 2.0237],</span>
<span class="go"> [ 2.0237, 6.6593]],</span>
<span class="go"> [[ 0.4187, 0.1830],</span>
<span class="go"> [ 0.1830, 0.1018]],</span>
<span class="go"> [[ 1.9348, -2.5744],</span>
<span class="go"> [-2.5744, 4.6386]]], dtype=torch.float64)</span>
<span class="gp">>>> </span><span class="n">l</span>
<span class="go">tensor([[[ 1.0784, 0.0000],</span>
<span class="go"> [ 1.8766, 1.7713]],</span>
<span class="go"> [[ 0.6471, 0.0000],</span>
<span class="go"> [ 0.2829, 0.1477]],</span>
<span class="go"> [[ 1.3910, 0.0000],</span>
<span class="go"> [-1.8509, 1.1014]]], dtype=torch.float64)</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">l</span><span class="p">,</span> <span class="n">l</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</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.linalg.cond">
<code class="sig-prename descclassname">torch.linalg.</code><code class="sig-name descname">cond</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">p=None</em>, <em class="sig-param">*</em>, <em class="sig-param">out=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.linalg.cond" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the condition number of a matrix <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>, or of each matrix in
a batched <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>, using the matrix norm defined by <code class="xref py py-attr docutils literal notranslate"><span class="pre">p</span></code>.</p>
<p>For norms <cite>{‘fro’, ‘nuc’, inf, -inf, 1, -1}</cite> this is defined as the matrix norm of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>
times the matrix norm of the inverse of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> computed using <a class="reference internal" href="#torch.linalg.norm" title="torch.linalg.norm"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.norm()</span></code></a>. While
for norms <cite>{None, 2, -2}</cite> this is defined as the ratio between the largest and smallest singular
values computed using <a class="reference internal" href="#torch.linalg.svd" title="torch.linalg.svd"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.svd()</span></code></a>.</p>
<p>This function supports float, double, cfloat and cdouble dtypes.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>When given inputs on a CUDA device, this function may synchronize that device with the CPU depending
on which norm <code class="xref py py-attr docutils literal notranslate"><span class="pre">p</span></code> is used.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>For norms <cite>{None, 2, -2}</cite>, <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> may be a non-square matrix or batch of non-square matrices.
For other norms, however, <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> must be a square matrix or a batch of square matrices,
and if this requirement is not satisfied a RuntimeError will be thrown.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>For norms <cite>{‘fro’, ‘nuc’, inf, -inf, 1, -1}</cite> if <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is a non-invertible matrix then
a tensor containing infinity will be returned. If <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is a batch of matrices and one
or more of them is not invertible then a RuntimeError will be thrown.</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 matrix of size <cite>(m, n)</cite> or the batch of matrices of size <cite>(*, m, n)</cite>
where <cite>*</cite> is one or more batch dimensions.</p></li>
<li><p><strong>p</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.9)"><em>float</em></a><em>, </em><em>inf</em><em>, </em><em>-inf</em><em>, </em><em>'fro'</em><em>, </em><em>'nuc'</em><em>, </em><em>optional</em>) – <p>the type of the matrix norm to use in the computations.
inf refers to <code class="xref py py-attr docutils literal notranslate"><span class="pre">float('inf')</span></code>, numpy’s <code class="xref py py-attr docutils literal notranslate"><span class="pre">inf</span></code> object, or any equivalent object.
The following norms can be used:</p>
<table class="docutils colwidths-auto align-default">
<thead>
<tr class="row-odd"><th class="head"><p>p</p></th>
<th class="head"><p>norm for matrices</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>None</p></td>
<td><p>ratio of the largest singular value to the smallest singular value</p></td>
</tr>
<tr class="row-odd"><td><p>’fro’</p></td>
<td><p>Frobenius norm</p></td>
</tr>
<tr class="row-even"><td><p>’nuc’</p></td>
<td><p>nuclear norm</p></td>
</tr>
<tr class="row-odd"><td><p>inf</p></td>
<td><p>max(sum(abs(x), dim=1))</p></td>
</tr>
<tr class="row-even"><td><p>-inf</p></td>
<td><p>min(sum(abs(x), dim=1))</p></td>
</tr>
<tr class="row-odd"><td><p>1</p></td>
<td><p>max(sum(abs(x), dim=0))</p></td>
</tr>
<tr class="row-even"><td><p>-1</p></td>
<td><p>min(sum(abs(x), dim=0))</p></td>
</tr>
<tr class="row-odd"><td><p>2</p></td>
<td><p>ratio of the largest singular value to the smallest singular value</p></td>
</tr>
<tr class="row-even"><td><p>-2</p></td>
<td><p>ratio of the smallest singular value to the largest singular value</p></td>
</tr>
</tbody>
</table>
<p>Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p>
</p></li>
</ul>
</dd>
<dt class="field-even">Keyword Arguments</dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – tensor to write the output to. Default is <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p>
</dd>
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><p>The condition number of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>. The output dtype is always real valued
even for complex inputs (e.g. float if <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is cfloat).</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">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</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">complex64</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cond</span><span class="p">(</span><span class="n">a</span><span class="p">)</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">tensor</span><span class="p">([[</span><span class="mf">1.</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cond</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">tensor([1.4142])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cond</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s1">'fro'</span><span class="p">)</span>
<span class="go">tensor(3.1623)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cond</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s1">'nuc'</span><span class="p">)</span>
<span class="go">tensor(9.2426)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cond</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="nb">float</span><span class="p">(</span><span class="s1">'inf'</span><span class="p">))</span>
<span class="go">tensor(2.)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cond</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="nb">float</span><span class="p">(</span><span class="s1">'-inf'</span><span class="p">))</span>
<span class="go">tensor(1.)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cond</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">tensor(2.)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cond</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">tensor(1.)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cond</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="go">tensor([1.4142])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cond</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>
<span class="go">tensor([0.7071])</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">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">tensor([[[-0.9204, 1.1140, 1.2055],</span>
<span class="go"> [ 0.3988, -0.2395, -0.7441],</span>
<span class="go"> [-0.5160, 0.3115, 0.2619]],</span>
<span class="go"> [[-2.2128, 0.9241, 2.1492],</span>
<span class="go"> [-1.1277, 2.7604, -0.8760],</span>
<span class="go"> [ 1.2159, 0.5960, 0.0498]]])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cond</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">tensor([[9.5917],</span>
<span class="go"> [3.2538]])</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">2</span><span class="p">,</span> <span class="mi">3</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">complex64</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">tensor([[[-0.4671-0.2137j, -0.1334-0.9508j, 0.6252+0.1759j],</span>
<span class="go"> [-0.3486-0.2991j, -0.1317+0.1252j, 0.3025-0.1604j],</span>
<span class="go"> [-0.5634+0.8582j, 0.1118-0.4677j, -0.1121+0.7574j]],</span>
<span class="go"> [[ 0.3964+0.2533j, 0.9385-0.6417j, -0.0283-0.8673j],</span>
<span class="go"> [ 0.2635+0.2323j, -0.8929-1.1269j, 0.3332+0.0733j],</span>
<span class="go"> [ 0.1151+0.1644j, -1.1163+0.3471j, -0.5870+0.1629j]]])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cond</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">tensor([[4.6245],</span>
<span class="go"> [4.5671]])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cond</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">tensor([9.2589, 9.3486])</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.linalg.det">
<code class="sig-prename descclassname">torch.linalg.</code><code class="sig-name descname">det</code><span class="sig-paren">(</span><em class="sig-param">input</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.linalg.det" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the determinant of a square matrix <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>, or of each square matrix
in a batched <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<p>This function supports float, double, cfloat and cdouble dtypes.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>When given inputs on a CUDA device, this function synchronizes that device with the CPU.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The determinant is computed using LU factorization. LAPACK’s <cite>getrf</cite> is used for CPU inputs,
and MAGMA’s <cite>getrf</cite> is used for CUDA inputs.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Backward through <cite>det</cite> internally uses <a class="reference internal" href="#torch.linalg.svd" title="torch.linalg.svd"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.svd()</span></code></a> when <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is not
invertible. In this case, double backward through <cite>det</cite> will be unstable when <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>
doesn’t have distinct singular values. See <a class="reference internal" href="#torch.linalg.svd" title="torch.linalg.svd"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.svd()</span></code></a> for more details.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input matrix of size <cite>(n, n)</cite> or the batch of matrices of size
<cite>(*, n, n)</cite> where <cite>*</cite> is one or more batch dimensions.</p>
</dd>
</dl>
<p>Example:</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">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">tensor([[ 0.9478, 0.9158, -1.1295],</span>
<span class="go"> [ 0.9701, 0.7346, -1.8044],</span>
<span class="go"> [-0.2337, 0.0557, 0.6929]])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">tensor(0.0934)</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">3</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</span>
<span class="go">tensor([[[ 0.9254, -0.6213],</span>
<span class="go"> [-0.5787, 1.6843]],</span>
<span class="go"> [[ 0.3242, -0.9665],</span>
<span class="go"> [ 0.4539, -0.0887]],</span>
<span class="go"> [[ 1.1336, -0.4025],</span>
<span class="go"> [-0.7089, 0.9032]]])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">tensor([1.1990, 0.4099, 0.7386])</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.linalg.slogdet">
<code class="sig-prename descclassname">torch.linalg.</code><code class="sig-name descname">slogdet</code><span class="sig-paren">(</span><em class="sig-param">input</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.linalg.slogdet" title="Permalink to this definition">¶</a></dt>
<dd><p>Calculates the sign and natural logarithm of the absolute value of a square matrix’s determinant,
or of the absolute values of the determinants of a batch of square matrices <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.
The determinant can be computed with <code class="docutils literal notranslate"><span class="pre">sign</span> <span class="pre">*</span> <span class="pre">exp(logabsdet)</span></code>.</p>
<p>Supports input of float, double, cfloat and cdouble datatypes.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>When given inputs on a CUDA device, this function synchronizes that device with the CPU.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The determinant is computed using LU factorization. LAPACK’s <cite>getrf</cite> is used for CPU inputs,
and MAGMA’s <cite>getrf</cite> is used for CUDA inputs.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>For matrices that have zero determinant, this returns <code class="docutils literal notranslate"><span class="pre">(0,</span> <span class="pre">-inf)</span></code>.
If <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is batched then the entries in the result tensors corresponding to matrices with
the zero determinant have sign 0 and the natural logarithm of the absolute value of the determinant -inf.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>input</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input matrix 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><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 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> or the batch of matrices 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 <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.46528em;vertical-align:0em;"></span><span class="mord">∗</span></span></span></span>
</span> is one or more batch dimensions.</p>
</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>) – tuple of two tensors to write the output to.</p>
</dd>
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><p>A namedtuple (sign, logabsdet) containing the sign of the determinant and the natural logarithm
of the absolute value of determinant, respectively.</p>
</dd>
</dl>
<p>Example:</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">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">A</span>
<span class="go">tensor([[ 0.0032, -0.2239, -1.1219],</span>
<span class="go"> [-0.6690, 0.1161, 0.4053],</span>
<span class="go"> [-1.6218, -0.9273, -0.0082]])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
<span class="go">tensor(-0.7576)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">logdet</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
<span class="go">tensor(nan)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">slogdet</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
<span class="go">torch.return_types.linalg_slogdet(sign=tensor(-1.), logabsdet=tensor(-0.2776))</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.linalg.eigh">
<code class="sig-prename descclassname">torch.linalg.</code><code class="sig-name descname">eigh</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">UPLO='L'</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.linalg.eigh" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the eigenvalues and eigenvectors of a complex Hermitian (or real symmetric)
matrix <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>, or of each such matrix in a batched <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<p>For a single matrix <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>, the tensor of eigenvalues <cite>w</cite> and the tensor of eigenvectors
<cite>V</cite> decompose the <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> such that <cite>input = V diag(w) Vᴴ</cite>, where <cite>Vᴴ</cite> is the transpose of <cite>V</cite>
for real-valued <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>, or the conjugate transpose of <cite>V</cite> for complex-valued <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<p>Since the matrix or matrices in <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> are assumed to be Hermitian, the imaginary part of their diagonals
is always treated as zero. When <code class="xref py py-attr docutils literal notranslate"><span class="pre">UPLO</span></code> is “L”, its default value, only the lower triangular part of each
matrix is used in the computation. When <code class="xref py py-attr docutils literal notranslate"><span class="pre">UPLO</span></code> is “U” only the upper triangular part of each matrix is used.</p>
<p>Supports input of float, double, cfloat and cdouble dtypes.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>When given inputs on a CUDA device, this function synchronizes that device with the CPU.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The eigenvalues/eigenvectors are computed using LAPACK’s <cite>syevd</cite> and <cite>heevd</cite> routines for CPU inputs,
and MAGMA’s <cite>syevd</cite> and <cite>heevd</cite> routines for CUDA inputs.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The eigenvalues of real symmetric or complex Hermitian matrices are always real.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The eigenvectors of matrices are not unique, so any eigenvector multiplied by a constant remains
a valid eigenvector. This function may compute different eigenvector representations on
different device types. Usually the difference is only in the sign of the eigenvector.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>See <a class="reference internal" href="#torch.linalg.eigvalsh" title="torch.linalg.eigvalsh"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.eigvalsh()</span></code></a> for a related function that computes only eigenvalues.
However, that function is not differentiable.</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 Hermitian <cite>n times n</cite> matrix or the batch of such matrices of size
<cite>(*, n, n)</cite> where <cite>*</cite> is one or more batch dimensions.</p></li>
<li><p><strong>UPLO</strong> (<em>'L'</em><em>, </em><em>'U'</em><em>, </em><em>optional</em>) – controls whether to use the upper-triangular or the lower-triangular part
of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> in the computations. Default is <code class="docutils literal notranslate"><span class="pre">'L'</span></code>.</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>) – tuple of two tensors to write the output to. Default is <code class="docutils literal notranslate"><span class="pre">None</span></code>.</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><dl class="simple">
<dt><strong>eigenvalues</strong> (<em>Tensor</em>): Shape <cite>(*, m)</cite>.</dt><dd><p>The eigenvalues in ascending order.</p>
</dd>
</dl>
</li>
<li><dl class="simple">
<dt><strong>eigenvectors</strong> (<em>Tensor</em>): Shape <cite>(*, m, m)</cite>.</dt><dd><p>The orthonormal eigenvectors of the <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
</dd>
</dl>
</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">2</span><span class="p">,</span> <span class="mi">2</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">complex128</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="o">.</span><span class="n">conj</span><span class="p">()</span> <span class="c1"># creates a Hermitian matrix</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">tensor([[2.9228+0.0000j, 0.2029-0.0862j],</span>
<span class="go"> [0.2029+0.0862j, 0.3464+0.0000j]], dtype=torch.complex128)</span>
<span class="gp">>>> </span><span class="n">w</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">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">w</span>
<span class="go">tensor([0.3277, 2.9415], dtype=torch.float64)</span>
<span class="gp">>>> </span><span class="n">v</span>
<span class="go">tensor([[-0.0846+-0.0000j, -0.9964+0.0000j],</span>
<span class="go"> [ 0.9170+0.3898j, -0.0779-0.0331j]], dtype=torch.complex128)</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">w</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</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">t</span><span class="p">()</span><span class="o">.</span><span class="n">conj</span><span class="p">())),</span> <span class="n">a</span><span class="p">)</span>
<span class="go">True</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">3</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="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="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">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"># creates a symmetric matrix</span>
<span class="gp">>>> </span><span class="n">w</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">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">a</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">w</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</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.linalg.eigvalsh">
<code class="sig-prename descclassname">torch.linalg.</code><code class="sig-name descname">eigvalsh</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">UPLO='L'</em>, <em class="sig-param">*</em>, <em class="sig-param">out=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.linalg.eigvalsh" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the eigenvalues of a complex Hermitian (or real symmetric) matrix <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>,
or of each such matrix in a batched <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>. The eigenvalues are returned in ascending order.</p>
<p>Since the matrix or matrices in <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> are assumed to be Hermitian, the imaginary part of their diagonals
is always treated as zero. When <code class="xref py py-attr docutils literal notranslate"><span class="pre">UPLO</span></code> is “L”, its default value, only the lower triangular part of
each matrix is used in the computation. When <code class="xref py py-attr docutils literal notranslate"><span class="pre">UPLO</span></code> is “U” only the upper triangular part of each matrix is used.</p>
<p>Supports input of float, double, cfloat and cdouble dtypes.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>When given inputs on a CUDA device, this function synchronizes that device with the CPU.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The eigenvalues are computed using LAPACK’s <cite>syevd</cite> and <cite>heevd</cite> routines for CPU inputs,
and MAGMA’s <cite>syevd</cite> and <cite>heevd</cite> routines for CUDA inputs.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The eigenvalues of real symmetric or complex Hermitian matrices are always real.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function doesn’t support backpropagation, please use <a class="reference internal" href="#torch.linalg.eigh" title="torch.linalg.eigh"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.eigh()</span></code></a> instead,
which also computes the eigenvectors.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>See <a class="reference internal" href="#torch.linalg.eigh" title="torch.linalg.eigh"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.eigh()</span></code></a> for a related function that computes both eigenvalues and eigenvectors.</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 Hermitian <cite>n times n</cite> matrix or the batch
of such matrices of size <cite>(*, n, n)</cite> where <cite>*</cite> is one or more batch dimensions.</p></li>
<li><p><strong>UPLO</strong> (<em>'L'</em><em>, </em><em>'U'</em><em>, </em><em>optional</em>) – controls whether to use the upper-triangular or the lower-triangular part
of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> in the computations. Default is <code class="docutils literal notranslate"><span class="pre">'L'</span></code>.</p></li>
</ul>
</dd>
<dt class="field-even">Keyword Arguments</dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – tensor to write the output to. Default is <code class="docutils literal notranslate"><span class="pre">None</span></code>.</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">2</span><span class="p">,</span> <span class="mi">2</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">complex128</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="o">.</span><span class="n">conj</span><span class="p">()</span> <span class="c1"># creates a Hermitian matrix</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">tensor([[2.9228+0.0000j, 0.2029-0.0862j],</span>
<span class="go"> [0.2029+0.0862j, 0.3464+0.0000j]], dtype=torch.complex128)</span>
<span class="gp">>>> </span><span class="n">w</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">eigvalsh</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">w</span>
<span class="go">tensor([0.3277, 2.9415], dtype=torch.float64)</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">3</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="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="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">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"># creates a symmetric matrix</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">tensor([[[ 2.8050, -0.3850],</span>
<span class="go"> [-0.3850, 3.2376]],</span>
<span class="go"> [[-1.0307, -2.7457],</span>
<span class="go"> [-2.7457, -1.7517]],</span>
<span class="go"> [[ 1.7166, 2.2207],</span>
<span class="go"> [ 2.2207, -2.0898]]], dtype=torch.float64)</span>
<span class="gp">>>> </span><span class="n">w</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">eigvalsh</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">w</span>
<span class="go">tensor([[ 2.5797, 3.4629],</span>
<span class="go"> [-4.1605, 1.3780],</span>
<span class="go"> [-3.1113, 2.7381]], dtype=torch.float64)</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.linalg.matrix_rank">
<code class="sig-prename descclassname">torch.linalg.</code><code class="sig-name descname">matrix_rank</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">tol=None</em>, <em class="sig-param">hermitian=False</em>, <em class="sig-param">*</em>, <em class="sig-param">out=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.linalg.matrix_rank" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the numerical rank of a matrix <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>, or of each matrix in a batched <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>.</p>
<p>The matrix rank is computed as the number of singular values (or absolute eigenvalues when <code class="xref py py-attr docutils literal notranslate"><span class="pre">hermitian</span></code> is <code class="docutils literal notranslate"><span class="pre">True</span></code>)
that are greater than the specified <code class="xref py py-attr docutils literal notranslate"><span class="pre">tol</span></code> threshold.</p>
<p>If <code class="xref py py-attr docutils literal notranslate"><span class="pre">tol</span></code> is not specified, <code class="xref py py-attr docutils literal notranslate"><span class="pre">tol</span></code> is set to <code class="docutils literal notranslate"><span class="pre">S.max(dim=-1)*max(input.shape[-2:])*eps</span></code>,
where <code class="docutils literal notranslate"><span class="pre">S</span></code> is the singular values (or absolute eigenvalues when <code class="xref py py-attr docutils literal notranslate"><span class="pre">hermitian</span></code> is <code class="docutils literal notranslate"><span class="pre">True</span></code>), and
<code class="docutils literal notranslate"><span class="pre">eps</span></code> is the epsilon value for the datatype of <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code>. The epsilon value can be obtained using
the <code class="docutils literal notranslate"><span class="pre">eps</span></code> attribute of <code class="xref py py-class docutils literal notranslate"><span class="pre">torch.finfo</span></code>.</p>
<p>Supports input of float, double, cfloat and cdouble dtypes.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>When given inputs on a CUDA device, this function synchronizes that device with the CPU.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The matrix rank is computed using singular value decomposition (see <a class="reference internal" href="#torch.linalg.svd" title="torch.linalg.svd"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.svd()</span></code></a>) by default.
If <code class="xref py py-attr docutils literal notranslate"><span class="pre">hermitian</span></code> is <code class="docutils literal notranslate"><span class="pre">True</span></code>, then <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is assumed to be Hermitian (symmetric if real-valued),
and the computation is done by obtaining the eigenvalues (see <a class="reference internal" href="#torch.linalg.eigvalsh" title="torch.linalg.eigvalsh"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.eigvalsh()</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 input matrix of size <cite>(m, n)</cite> or the batch of matrices of size <cite>(*, m, n)</cite>
where <cite>*</cite> is one or more batch dimensions.</p></li>
<li><p><strong>tol</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.9)"><em>float</em></a><em>, </em><em>optional</em>) – the tolerance value. Default is <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
<li><p><strong>hermitian</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.9)"><em>bool</em></a><em>, </em><em>optional</em>) – indicates whether <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is Hermitian. Default is <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
</ul>
</dd>
<dt class="field-even">Keyword Arguments</dt>
<dd class="field-even"><p><strong>out</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – tensor to write the output to. Default is <code class="docutils literal notranslate"><span class="pre">None</span></code>.</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">eye</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_rank</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">tensor(10)</span>
<span class="gp">>>> </span><span class="n">b</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_rank</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="go">tensor(9)</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">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_rank</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">tensor([2, 2, 2, 2])</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">2</span><span class="p">,</span> <span class="mi">4</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="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_rank</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">tensor([[2, 2, 2, 2],</span>
<span class="go"> [2, 2, 2, 2]])</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">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</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">complex64</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_rank</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">tensor([[3, 3, 3, 3],</span>
<span class="go"> [3, 3, 3, 3]])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_rank</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">hermitian</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="go">tensor([[3, 3, 3, 3],</span>
<span class="go"> [3, 3, 3, 3]])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_rank</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1.0</span><span class="p">)</span>
<span class="go">tensor([[3, 2, 2, 2],</span>
<span class="go"> [1, 2, 1, 2]])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">matrix_rank</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">hermitian</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="go">tensor([[2, 2, 2, 1],</span>
<span class="go"> [1, 2, 2, 2]])</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.linalg.norm">
<code class="sig-prename descclassname">torch.linalg.</code><code class="sig-name descname">norm</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">ord=None</em>, <em class="sig-param">dim=None</em>, <em class="sig-param">keepdim=False</em>, <em class="sig-param">*</em>, <em class="sig-param">out=None</em>, <em class="sig-param">dtype=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.linalg.norm" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the matrix norm or vector norm of a given tensor.</p>
<p>This function can calculate one of eight different types of matrix norms, or one
of an infinite number of vector norms, depending on both the number of reduction
dimensions and the value of the <cite>ord</cite> parameter.</p>
<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. If dim is None, x must be 1-D or 2-D, unless <code class="xref py py-attr docutils literal notranslate"><span class="pre">ord</span></code>
is None. If both <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim</span></code> and <code class="xref py py-attr docutils literal notranslate"><span class="pre">ord</span></code> are None, the 2-norm of the input flattened to 1-D
will be returned. Its data type must be either a floating point or complex type. For complex
inputs, the norm is calculated on of the absolute values of each element. If the input is
complex and neither <code class="xref py py-attr docutils literal notranslate"><span class="pre">dtype</span></code> nor <code class="xref py py-attr docutils literal notranslate"><span class="pre">out</span></code> is specified, the result’s data type will
be the corresponding floating point type (e.g. float if <code class="xref py py-attr docutils literal notranslate"><span class="pre">input</span></code> is complexfloat).</p></li>
<li><p><strong>ord</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.9)"><em>float</em></a><em>, </em><em>inf</em><em>, </em><em>-inf</em><em>, </em><em>'fro'</em><em>, </em><em>'nuc'</em><em>, </em><em>optional</em>) – <p>The order of norm.
inf refers to <code class="xref py py-attr docutils literal notranslate"><span class="pre">float('inf')</span></code>, numpy’s <code class="xref py py-attr docutils literal notranslate"><span class="pre">inf</span></code> object, or any equivalent object.
The following norms can be calculated:</p>
<table class="docutils colwidths-auto align-default">
<thead>
<tr class="row-odd"><th class="head"><p>ord</p></th>
<th class="head"><p>norm for matrices</p></th>
<th class="head"><p>norm for vectors</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>None</p></td>
<td><p>Frobenius norm</p></td>
<td><p>2-norm</p></td>
</tr>
<tr class="row-odd"><td><p>’fro’</p></td>
<td><p>Frobenius norm</p></td>
<td><p>– not supported –</p></td>
</tr>
<tr class="row-even"><td><p>‘nuc’</p></td>
<td><p>nuclear norm</p></td>
<td><p>– not supported –</p></td>
</tr>
<tr class="row-odd"><td><p>inf</p></td>
<td><p>max(sum(abs(x), dim=1))</p></td>
<td><p>max(abs(x))</p></td>
</tr>
<tr class="row-even"><td><p>-inf</p></td>
<td><p>min(sum(abs(x), dim=1))</p></td>
<td><p>min(abs(x))</p></td>