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<h1>torch.linalg<a class="headerlink" href="#torch-linalg" title="Permalink to this headline">¶</a></h1>
<p>Common linear algebra operations.</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.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>Alias of <a class="reference internal" href="generated/torch.det.html#torch.det" title="torch.det"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.det()</span></code></a>.</p>
</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.</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>
</tr>
<tr class="row-odd"><td><p>0</p></td>
<td><p>– not supported –</p></td>
<td><p>sum(x != 0)</p></td>
</tr>
<tr class="row-even"><td><p>1</p></td>
<td><p>max(sum(abs(x), dim=0))</p></td>
<td><p>as below</p></td>
</tr>
<tr class="row-odd"><td><p>-1</p></td>
<td><p>min(sum(abs(x), dim=0))</p></td>
<td><p>as below</p></td>
</tr>
<tr class="row-even"><td><p>2</p></td>
<td><p>2-norm (largest sing. value)</p></td>
<td><p>as below</p></td>
</tr>
<tr class="row-odd"><td><p>-2</p></td>
<td><p>smallest singular value</p></td>
<td><p>as below</p></td>
</tr>
<tr class="row-even"><td><p>other</p></td>
<td><p>– not supported –</p></td>
<td><p>sum(abs(x)**ord)**(1./ord)</p></td>
</tr>
</tbody>
</table>
<p>Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p>
</p></li>
<li><p><strong>dim</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><em>2-tuple of python:ints</em><em>, </em><em>2-list of python:ints</em><em>, </em><em>optional</em>) – If <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim</span></code> is an int,
vector norm will be calculated over the specified dimension. If <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim</span></code>
is a 2-tuple of ints, matrix norm will be calculated over the specified
dimensions. If <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim</span></code> is None, matrix norm will be calculated
when the input tensor has two dimensions, and vector norm will be
calculated when the input tensor has one dimension. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
<li><p><strong>keepdim</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>) – If set to True, the reduced dimensions are retained
in the result as dimensions with size one. Default: <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"><ul class="simple">
<li><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></li>
<li><p><strong>dtype</strong> (<code class="xref py py-class docutils literal notranslate"><span class="pre">torch.dtype</span></code>, optional) – If specified, the input tensor is cast to
<code class="xref py py-attr docutils literal notranslate"><span class="pre">dtype</span></code> before performing the operation, and the returned tensor’s type
will be <code class="xref py py-attr docutils literal notranslate"><span class="pre">dtype</span></code>. If this argument is used in conjunction with the
<code class="xref py py-attr docutils literal notranslate"><span class="pre">out</span></code> argument, the output tensor’s type must match this argument or a
RuntimeError will be raised. This argument is not currently supported for
<code class="xref py py-attr docutils literal notranslate"><span class="pre">ord='nuc'</span></code> or <code class="xref py py-attr docutils literal notranslate"><span class="pre">ord='fro'</span></code>. Default: <code class="docutils literal notranslate"><span class="pre">None</span></code></p></li>
</ul>
</dd>
</dl>
<p>Examples:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">torch</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">linalg</span> <span class="k">as</span> <span class="n">LA</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">arange</span><span class="p">(</span><span class="mi">9</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">float</span><span class="p">)</span> <span class="o">-</span> <span class="mi">4</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">tensor([-4., -3., -2., -1., 0., 1., 2., 3., 4.])</span>
<span class="gp">>>> </span><span class="n">b</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">reshape</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">b</span>
<span class="go">tensor([[-4., -3., -2.],</span>
<span class="go"> [-1., 0., 1.],</span>
<span class="go"> [ 2., 3., 4.]])</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">tensor(7.7460)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="go">tensor(7.7460)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="s1">'fro'</span><span class="p">)</span>
<span class="go">tensor(7.7460)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</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(4.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">b</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(9.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="nb">float</span><span class="p">(</span><span class="s1">'inf'</span><span class="p">))</span>
<span class="go">tensor(0.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="o">-</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">LA</span><span class="o">.</span><span class="n">norm</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(20.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">tensor(7.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</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(0.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">tensor(6.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</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(7.7460)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="go">tensor(7.3485)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</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.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">double</span><span class="p">(),</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>
<span class="go">tensor(1.8570e-16, dtype=torch.float64)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="go">tensor(5.8480)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">)</span>
<span class="go">tensor(0.)</span>
</pre></div>
</div>
<p>Using the <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim</span></code> argument to compute vector norms:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">c</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="mf">2.</span><span class="p">,</span> <span class="mf">3.</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="go">tensor([1.4142, 2.2361, 5.0000])</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="go">tensor([3.7417, 4.2426])</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="nb">ord</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="go">tensor([6., 6.])</span>
</pre></div>
</div>
<p>Using the <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim</span></code> argument to compute matrix norms:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">m</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">8</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">float</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</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="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</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="go">tensor([ 3.7417, 11.2250])</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">m</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:]),</span> <span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">m</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:])</span>
<span class="go">(tensor(3.7417), tensor(11.2250))</span>
</pre></div>
</div>
</dd></dl>
</div>
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