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332 | 332 | <h1 id="extensions-generated-array-api-linalg-eigh--page-root">eigh<a class="headerlink" href="#extensions-generated-array-api-linalg-eigh--page-root" title="Link to this heading">¶</a></h1> |
333 | 333 | <dl class="py function"> |
334 | 334 | <dt class="sig sig-object py" id="array_api.linalg.eigh"> |
335 | | -<span class="sig-name descname"><span class="pre">eigh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">array</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">/</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><span class="pre">Tuple</span><span class="p"><span class="pre">[</span></span><span class="pre">array</span><span class="p"><span class="pre">]</span></span></span></span><a class="headerlink" href="#array_api.linalg.eigh" title="Link to this definition">¶</a></dt> |
| 335 | +<span class="sig-name descname"><span class="pre">eigh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">array</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">/</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">→</span> <span class="sig-return-typehint"><span class="pre">Tuple</span><span class="p"><span class="pre">[</span></span><span class="pre">array</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">array</span><span class="p"><span class="pre">]</span></span></span></span><a class="headerlink" href="#array_api.linalg.eigh" title="Link to this definition">¶</a></dt> |
336 | 336 | <dd><p>Returns an eigenvalue decomposition of a complex Hermitian or real symmetric matrix (or a stack of matrices) <code class="docutils literal notranslate"><span class="pre">x</span></code>.</p> |
337 | 337 | <p>If <code class="docutils literal notranslate"><span class="pre">x</span></code> is real-valued, let <span class="math notranslate nohighlight">\(\mathbb{K}\)</span> be the set of real numbers <span class="math notranslate nohighlight">\(\mathbb{R}\)</span>, and, if <code class="docutils literal notranslate"><span class="pre">x</span></code> is complex-valued, let <span class="math notranslate nohighlight">\(\mathbb{K}\)</span> be the set of complex numbers <span class="math notranslate nohighlight">\(\mathbb{C}\)</span>.</p> |
338 | 338 | <p>The <strong>eigenvalue decomposition</strong> of a complex Hermitian or real symmetric matrix <span class="math notranslate nohighlight">\(x \in\ \mathbb{K}^{n \times n}\)</span> is defined as</p> |
@@ -361,7 +361,7 @@ <h1 id="extensions-generated-array-api-linalg-eigh--page-root">eigh<a class="hea |
361 | 361 | <dd class="field-odd"><p><strong>x</strong> (<em>array</em>) – input array having shape <code class="docutils literal notranslate"><span class="pre">(...,</span> <span class="pre">M,</span> <span class="pre">M)</span></code> and whose innermost two dimensions form square matrices. Should have a floating-point data type.</p> |
362 | 362 | </dd> |
363 | 363 | <dt class="field-even">Returns<span class="colon">:</span></dt> |
364 | | -<dd class="field-even"><p><strong>out</strong> (<em>Tuple[array]</em>) – a namedtuple (<code class="docutils literal notranslate"><span class="pre">eigenvalues</span></code>, <code class="docutils literal notranslate"><span class="pre">eigenvectors</span></code>) whose</p> |
| 364 | +<dd class="field-even"><p><strong>out</strong> (<em>Tuple[array, array]</em>) – a namedtuple (<code class="docutils literal notranslate"><span class="pre">eigenvalues</span></code>, <code class="docutils literal notranslate"><span class="pre">eigenvectors</span></code>) whose</p> |
365 | 365 | <ul class="simple"> |
366 | 366 | <li><p>first element must have the field name <code class="docutils literal notranslate"><span class="pre">eigenvalues</span></code> (corresponding to <span class="math notranslate nohighlight">\(\operatorname{diag}\Lambda\)</span> above) and must be an array consisting of computed eigenvalues. The array containing the eigenvalues must have shape <code class="docutils literal notranslate"><span class="pre">(...,</span> <span class="pre">M)</span></code> and must have a real-valued floating-point data type whose precision matches the precision of <code class="docutils literal notranslate"><span class="pre">x</span></code> (e.g., if <code class="docutils literal notranslate"><span class="pre">x</span></code> is <code class="docutils literal notranslate"><span class="pre">complex128</span></code>, then <code class="docutils literal notranslate"><span class="pre">eigenvalues</span></code> must be <code class="docutils literal notranslate"><span class="pre">float64</span></code>).</p></li> |
367 | 367 | <li><p>second element have have the field name <code class="docutils literal notranslate"><span class="pre">eigenvectors</span></code> (corresponding to <span class="math notranslate nohighlight">\(Q\)</span> above) and must be an array where the columns of the inner most matrices contain the computed eigenvectors. These matrices must be orthogonal. The array containing the eigenvectors must have shape <code class="docutils literal notranslate"><span class="pre">(...,</span> <span class="pre">M,</span> <span class="pre">M)</span></code> and must have the same data type as <code class="docutils literal notranslate"><span class="pre">x</span></code>.</p></li> |
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