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 \bigr\}.\]</p><p>This manifold is modelled as a submanifold of <a href="symmetricpositivedefinite.html#Manifolds.SymmetricPositiveDefinite"><code>SymmetricPositiveDefinite</code></a><code>(n)</code>.</p><p>These matrices are sometimes also called <a href="https://en.wiktionary.org/wiki/isochoric">isochoric</a>, which refers to the interpretation of the matrix representing an ellipsoid. All ellipsoids that represent points on this manifold have the same volume.</p><p>The tangent space is modelled the same as for <a href="symmetricpositivedefinite.html#Manifolds.SymmetricPositiveDefinite"><code>SymmetricPositiveDefinite</code></a><code>(n)</code> and consists of all symmetric matrices with zero trace</p><p class="math-container">\[    T_p\mathcal P_d(n) =
     \bigl\{
         X \in \mathbb R^{n×n} \big|\ X=X^\mathrm{T} \text{ and } \operatorname{tr}(p) = 0
-    \bigr\},\]</p><p>since for a constant determinant we require that <code>0 = D\det(p)[Z] = \det(p)\operatorname{tr}(p^{-1}Z)</code> for all tangent vectors <span>$Z$</span>. Additionally we store the tangent vectors as <code>X=p^{-1}Z</code>, i.e. symmetric matrices.</p><p><strong>Constructor</strong></p><pre><code class="nohighlight hljs">SPDFixedDeterminant(n::Int, d::Real=1.0; parameter::Symbol=:type)</code></pre><p>Generate the manifold <span>$\mathcal P_d(n) \subset \mathcal P(n)$</span> of determinant <span>$d$</span>, which defaults to 1.</p><p><code>parameter</code>: whether a type parameter should be used to store <code>n</code>. By default size is stored in type. Value can either be <code>:field</code> or <code>:type</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/2104befae9261a20cabb0994dc4060935ac17b62/src/manifolds/SPDFixedDeterminant.jl#L1-L39">source</a></section></article><p>This manifold can is a submanifold of the <a href="symmetricpositivedefinite.html#SymmetricPositiveDefiniteSection">symmetric positive definite matrices</a> and hence inherits most properties therefrom.</p><p>The differences are the functions</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="ManifoldsBase.check_point-Tuple{SPDFixedDeterminant, Any}" href="#ManifoldsBase.check_point-Tuple{SPDFixedDeterminant, Any}"><code>ManifoldsBase.check_point</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">check_point(M::SPDFixedDeterminant, p; kwargs...)</code></pre><p>Check whether <code>p</code> is a valid manifold point on the <a href="spdfixeddeterminant.html#Manifolds.SPDFixedDeterminant"><code>SPDFixedDeterminant</code></a><code>(n,d)</code> <code>M</code>, i.e. whether <code>p</code> is a <a href="symmetricpositivedefinite.html#Manifolds.SymmetricPositiveDefinite"><code>SymmetricPositiveDefinite</code></a> matrix of size <code>(n, n)</code></p><p>with determinant <span>$\det(p) =$</span><code>M.d</code>.</p><p>The tolerance for the determinant of <code>p</code> can be set using <code>kwargs...</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/2104befae9261a20cabb0994dc4060935ac17b62/src/manifolds/SPDFixedDeterminant.jl#L55-L64">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="ManifoldsBase.check_vector-Tuple{SPDFixedDeterminant, Any, Any}" href="#ManifoldsBase.check_vector-Tuple{SPDFixedDeterminant, Any, Any}"><code>ManifoldsBase.check_vector</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">check_vector(M::SPDFixedDeterminant, p, X; kwargs... )</code></pre><p>Check whether <code>X</code> is a tangent vector to manifold point <code>p</code> on the <a href="spdfixeddeterminant.html#Manifolds.SPDFixedDeterminant"><code>SPDFixedDeterminant</code></a> <code>M</code>, i.e. <code>X</code> has to be a tangent vector on <a href="symmetricpositivedefinite.html#Manifolds.SymmetricPositiveDefinite"><code>SymmetricPositiveDefinite</code></a>, so a symmetric matrix, and additionally fulfill <span>$\operatorname{tr}(X) = 0$</span>.</p><p>The tolerance for the trace check of <code>X</code> can be set using <code>kwargs...</code>, which influences the <code>isapprox</code>-check.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/2104befae9261a20cabb0994dc4060935ac17b62/src/manifolds/SPDFixedDeterminant.jl#L75-L84">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="ManifoldsBase.project-Tuple{SPDFixedDeterminant, Any, Any}" href="#ManifoldsBase.project-Tuple{SPDFixedDeterminant, Any, Any}"><code>ManifoldsBase.project</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">Y = project(M::SPDFixedDeterminant, p, X)
-project!(M::SPDFixedDeterminant, Y, p, X)</code></pre><p>Project the symmetric matrix <code>X</code> onto the tangent space at <code>p</code> of the (sub-)manifold of s.p.d. matrices of determinant <code>M.d</code> (in place of <code>Y</code>), by setting its diagonal (and hence its trace) to zero.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/2104befae9261a20cabb0994dc4060935ac17b62/src/manifolds/SPDFixedDeterminant.jl#L144-L152">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="ManifoldsBase.project-Tuple{SPDFixedDeterminant, Any}" href="#ManifoldsBase.project-Tuple{SPDFixedDeterminant, Any}"><code>ManifoldsBase.project</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">q = project(M::SPDFixedDeterminant, p)
-project!(M::SPDFixedDeterminant, q, p)</code></pre><p>Project the symmetric positive definite (s.p.d.) matrix <code>p</code> from the embedding onto the (sub-)manifold of s.p.d. matrices of determinant <code>M.d</code> (in place of <code>q</code>).</p><p>The formula reads</p><p class="math-container">\[q = \Bigl(\frac{d}{\det(p)}\Bigr)^{\frac{1}{n}}p\]</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/2104befae9261a20cabb0994dc4060935ac17b62/src/manifolds/SPDFixedDeterminant.jl#L123-L135">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="symmetricpositivedefinite.html">« Symmetric positive definite</a><a class="docs-footer-nextpage" href="symmetricpsdfixedrank.html">Symmetric positive semidefinite fixed rank »</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Wednesday 8 November 2023 19:06">Wednesday 8 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+    \bigr\},\]</p><p>since for a constant determinant we require that <code>0 = D\det(p)[Z] = \det(p)\operatorname{tr}(p^{-1}Z)</code> for all tangent vectors <span>$Z$</span>. Additionally we store the tangent vectors as <code>X=p^{-1}Z</code>, i.e. symmetric matrices.</p><p><strong>Constructor</strong></p><pre><code class="nohighlight hljs">SPDFixedDeterminant(n::Int, d::Real=1.0; parameter::Symbol=:type)</code></pre><p>Generate the manifold <span>$\mathcal P_d(n) \subset \mathcal P(n)$</span> of determinant <span>$d$</span>, which defaults to 1.</p><p><code>parameter</code>: whether a type parameter should be used to store <code>n</code>. By default size is stored in type. Value can either be <code>:field</code> or <code>:type</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/5d267d532036399ed9d0e31d19345dee10b64bdb/src/manifolds/SPDFixedDeterminant.jl#L1-L39">source</a></section></article><p>This manifold can is a submanifold of the <a href="symmetricpositivedefinite.html#SymmetricPositiveDefiniteSection">symmetric positive definite matrices</a> and hence inherits most properties therefrom.</p><p>The differences are the functions</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="ManifoldsBase.check_point-Tuple{SPDFixedDeterminant, Any}" href="#ManifoldsBase.check_point-Tuple{SPDFixedDeterminant, Any}"><code>ManifoldsBase.check_point</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">check_point(M::SPDFixedDeterminant, p; kwargs...)</code></pre><p>Check whether <code>p</code> is a valid manifold point on the <a href="spdfixeddeterminant.html#Manifolds.SPDFixedDeterminant"><code>SPDFixedDeterminant</code></a><code>(n,d)</code> <code>M</code>, i.e. whether <code>p</code> is a <a href="symmetricpositivedefinite.html#Manifolds.SymmetricPositiveDefinite"><code>SymmetricPositiveDefinite</code></a> matrix of size <code>(n, n)</code></p><p>with determinant <span>$\det(p) =$</span><code>M.d</code>.</p><p>The tolerance for the determinant of <code>p</code> can be set using <code>kwargs...</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/5d267d532036399ed9d0e31d19345dee10b64bdb/src/manifolds/SPDFixedDeterminant.jl#L55-L64">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="ManifoldsBase.check_vector-Tuple{SPDFixedDeterminant, Any, Any}" href="#ManifoldsBase.check_vector-Tuple{SPDFixedDeterminant, Any, Any}"><code>ManifoldsBase.check_vector</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">check_vector(M::SPDFixedDeterminant, p, X; kwargs... )</code></pre><p>Check whether <code>X</code> is a tangent vector to manifold point <code>p</code> on the <a href="spdfixeddeterminant.html#Manifolds.SPDFixedDeterminant"><code>SPDFixedDeterminant</code></a> <code>M</code>, i.e. <code>X</code> has to be a tangent vector on <a href="symmetricpositivedefinite.html#Manifolds.SymmetricPositiveDefinite"><code>SymmetricPositiveDefinite</code></a>, so a symmetric matrix, and additionally fulfill <span>$\operatorname{tr}(X) = 0$</span>.</p><p>The tolerance for the trace check of <code>X</code> can be set using <code>kwargs...</code>, which influences the <code>isapprox</code>-check.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/5d267d532036399ed9d0e31d19345dee10b64bdb/src/manifolds/SPDFixedDeterminant.jl#L75-L84">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="ManifoldsBase.project-Tuple{SPDFixedDeterminant, Any, Any}" href="#ManifoldsBase.project-Tuple{SPDFixedDeterminant, Any, Any}"><code>ManifoldsBase.project</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">Y = project(M::SPDFixedDeterminant, p, X)
+project!(M::SPDFixedDeterminant, Y, p, X)</code></pre><p>Project the symmetric matrix <code>X</code> onto the tangent space at <code>p</code> of the (sub-)manifold of s.p.d. matrices of determinant <code>M.d</code> (in place of <code>Y</code>), by setting its diagonal (and hence its trace) to zero.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/5d267d532036399ed9d0e31d19345dee10b64bdb/src/manifolds/SPDFixedDeterminant.jl#L144-L152">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="ManifoldsBase.project-Tuple{SPDFixedDeterminant, Any}" href="#ManifoldsBase.project-Tuple{SPDFixedDeterminant, Any}"><code>ManifoldsBase.project</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">q = project(M::SPDFixedDeterminant, p)
+project!(M::SPDFixedDeterminant, q, p)</code></pre><p>Project the symmetric positive definite (s.p.d.) matrix <code>p</code> from the embedding onto the (sub-)manifold of s.p.d. matrices of determinant <code>M.d</code> (in place of <code>q</code>).</p><p>The formula reads</p><p class="math-container">\[q = \Bigl(\frac{d}{\det(p)}\Bigr)^{\frac{1}{n}}p\]</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaManifolds/Manifolds.jl/blob/5d267d532036399ed9d0e31d19345dee10b64bdb/src/manifolds/SPDFixedDeterminant.jl#L123-L135">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="symmetricpositivedefinite.html">« Symmetric positive definite</a><a class="docs-footer-nextpage" href="symmetricpsdfixedrank.html">Symmetric positive semidefinite fixed rank »</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Thursday 9 November 2023 18:54">Thursday 9 November 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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