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class="is-hidden-tablet"><li class="is-active"><a href>Symmetric and Skew-Symmetric Matrices</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/JuliaGNI/GeometricMachineLearning.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/JuliaGNI/GeometricMachineLearning.jl/blob/main/docs/src/arrays/skew_symmetric_matrix.md#L" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="SymmetricMatrix-and-SkewSymMatrix"><a class="docs-heading-anchor" href="#SymmetricMatrix-and-SkewSymMatrix"><code>SymmetricMatrix</code> and <code>SkewSymMatrix</code></a><a id="SymmetricMatrix-and-SkewSymMatrix-1"></a><a class="docs-heading-anchor-permalink" href="#SymmetricMatrix-and-SkewSymMatrix" title="Permalink"></a></h1><p>There are special implementations of symmetric and skew-symmetric matrices in <code>GeometricMachineLearning.jl</code>. They are implemented to work on GPU and for multiplication with tensors. The following image demonstrates how the data necessary for an instance of <code>SkewSymMatrix</code> are stored<sup class="footnote-reference"><a id="citeref-1" href="#footnote-1">[1]</a></sup>:</p><object type="image/svg+xml" class="display-light-only" data=../../tikz/skew_sym_visualization.png></object><pre><code class="language-julia hljs">end #</code></pre><object type="image/svg+xml" class="display-dark-only" data=../../tikz/skew_sym_visualization_dark.png></object><p>So what is stored internally is a vector of size <span>$n(n-1)/2$</span> for the skew-symmetric matrix and a vector of size <span>$n(n+1)/2$</span> for the symmetric matrix. We can sample a random skew-symmetric matrix: </p><pre><code class="language-julia hljs">using GeometricMachineLearning # hide
+
+A = rand(SkewSymMatrix, 5)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">5×5 SkewSymMatrix{Float64, Vector{Float64}}:
+ 0.0       -0.593087  -0.153597  -0.232307  -0.265829
+ 0.593087   0.0       -0.711961  -0.861843  -0.799511
+ 0.153597   0.711961   0.0       -0.382979  -0.591081
+ 0.232307   0.861843   0.382979   0.0       -0.237153
+ 0.265829   0.799511   0.591081   0.237153   0.0</code></pre><p>and then access the vector:</p><pre><code class="language-julia hljs">A.S</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">10-element Vector{Float64}:
+ 0.5930865209768201
+ 0.15359678424579415
+ 0.7119606697584316
+ 0.23230691635266243
+ 0.8618433216036477
+ 0.38297891816347196
+ 0.265829288847496
+ 0.7995111947754676
+ 0.591080909034458
+ 0.23715282889218914</code></pre><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>It works similarly for <code>SymmetricMatrix</code>. </li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../manifolds/existence_and_uniqueness_theorem/">« Differential Equations and the EAU theorem</a><a class="docs-footer-nextpage" href="../stiefel_lie_alg_horizontal/">Stiefel Global Tangent Space »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.3.0 on <span class="colophon-date" title="Wednesday 10 April 2024 16:20">Wednesday 10 April 2024</span>. Using Julia version 1.10.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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