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a ⊙ b</code></pre><p>For arrays <code>a</code> and <code>b</code>, perform elementwise multiplication. <code>a</code> and <code>b</code> must have identical <code>axes</code>.</p><p><code>⊙</code> can be passed as an operator to higher-order functions.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl">julia&gt; a = [2, 3]; b = [5, 7];

julia&gt; a ⊙ b
2-element Array{Int64,1}:
2-element Vector{Int64}:
10
21

julia&gt; a ⊙ [5]
ERROR: DimensionMismatch(&quot;Axes of `A` and `B` must match, got (Base.OneTo(2),) and (Base.OneTo(1),)&quot;)
[...]</code></pre><p>See also <code>hadamard!(y, a, b)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/TensorCore.jl/blob/7e9ad327756cfe3ded97a1c21c5e52c93a1d4db5/src/TensorCore.jl#LL9-L33">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="TensorCore.hadamard!" href="#TensorCore.hadamard!"><code>TensorCore.hadamard!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">hadamard!(dest, A, B)</code></pre><p>Similar to <code>hadamard(A, B)</code> (which can also be written <code>A ⊙ B</code>), but stores its results in the pre-allocated array <code>dest</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/TensorCore.jl/blob/7e9ad327756cfe3ded97a1c21c5e52c93a1d4db5/src/TensorCore.jl#LL43-L48">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="TensorCore.tensor" href="#TensorCore.tensor"><code>TensorCore.tensor</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">tensor(A, B)
ERROR: DimensionMismatch: Axes of `A` and `B` must match, got (Base.OneTo(2),) and (Base.OneTo(1),)
[...]</code></pre><p>See also <code>hadamard!(y, a, b)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/TensorCore.jl/blob/83a840a962dcb64c64d9cb1346e82c83a4afaab1/src/TensorCore.jl#LL9-L33">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="TensorCore.hadamard!" href="#TensorCore.hadamard!"><code>TensorCore.hadamard!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">hadamard!(dest, A, B)</code></pre><p>Similar to <code>hadamard(A, B)</code> (which can also be written <code>A ⊙ B</code>), but stores its results in the pre-allocated array <code>dest</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/TensorCore.jl/blob/83a840a962dcb64c64d9cb1346e82c83a4afaab1/src/TensorCore.jl#LL43-L48">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="TensorCore.tensor" href="#TensorCore.tensor"><code>TensorCore.tensor</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">tensor(A, B)
A ⊗ B</code></pre><p>Compute the tensor product of <code>A</code> and <code>B</code>. If <code>C = A ⊗ B</code>, then <code>C[i1, ..., im, j1, ..., jn] = A[i1, ... im] * B[j1, ..., jn]</code>.</p><p>For vectors <code>v</code> and <code>w</code>, the Kronecker product is related to the tensor product by <code>kron(v,w) == vec(w ⊗ v)</code> or <code>w ⊗ v == reshape(kron(v,w), (length(w), length(v)))</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl">julia&gt; a = [2, 3]; b = [5, 7, 11];

julia&gt; a ⊗ b
2×3 Array{Int64,2}:
2×3 Matrix{Int64}:
10 14 22
15 21 33</code></pre><p>See also <code>tensor!(Y,A,B)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/TensorCore.jl/blob/7e9ad327756cfe3ded97a1c21c5e52c93a1d4db5/src/TensorCore.jl#LL62-L82">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="TensorCore.tensor!" href="#TensorCore.tensor!"><code>TensorCore.tensor!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">tensor!(dest, A, B)</code></pre><p>Similar to <code>tensor(A, B)</code> (which can also be written <code>A ⊗ B</code>), but stores its results in the pre-allocated array <code>dest</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/TensorCore.jl/blob/7e9ad327756cfe3ded97a1c21c5e52c93a1d4db5/src/TensorCore.jl#LL100-L105">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="TensorCore.boxdot" href="#TensorCore.boxdot"><code>TensorCore.boxdot</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">boxdot(A,B) = A ⊡ B # \boxdot</code></pre><p>Generalised matrix multiplication: Contracts the last dimension of <code>A</code> with the first dimension of <code>B</code>, for any <code>ndims(A)</code> &amp; <code>ndims(B)</code>. If both are vectors, then it returns a scalar <code>== sum(A .* B)</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl">julia&gt; A = rand(3,4,5); B = rand(5,6,7);
15 21 33</code></pre><p>See also <code>tensor!(Y,A,B)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/TensorCore.jl/blob/83a840a962dcb64c64d9cb1346e82c83a4afaab1/src/TensorCore.jl#LL62-L82">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="TensorCore.tensor!" href="#TensorCore.tensor!"><code>TensorCore.tensor!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">tensor!(dest, A, B)</code></pre><p>Similar to <code>tensor(A, B)</code> (which can also be written <code>A ⊗ B</code>), but stores its results in the pre-allocated array <code>dest</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/TensorCore.jl/blob/83a840a962dcb64c64d9cb1346e82c83a4afaab1/src/TensorCore.jl#LL100-L105">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="TensorCore.boxdot" href="#TensorCore.boxdot"><code>TensorCore.boxdot</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">boxdot(A,B) = A ⊡ B # \boxdot</code></pre><p>Generalised matrix multiplication: Contracts the last dimension of <code>A</code> with the first dimension of <code>B</code>, for any <code>ndims(A)</code> &amp; <code>ndims(B)</code>. If both are vectors, then it returns a scalar <code>== sum(A .* B)</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl">julia&gt; A = rand(3,4,5); B = rand(5,6,7);

julia&gt; size(A ⊡ B)
(3, 4, 6, 7)
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DimensionMismatch(&quot;neighbouring axes of `A` and `B` must match, got Base.OneTo(7) and Base.OneTo(3)&quot;)</code></pre><p>This is the same behaviour as Mathematica&#39;s function <code>Dot[A, B]</code>. It is not identicaly to Python&#39;s <code>numpy.dot(A, B)</code>, which contracts with the second-last dimension of <code>B</code> instead of the first, but both keep all the other dimensions. Unlike Julia&#39;s <code>LinearAlgebra.dot</code>, it does not conjugate <code>A</code>, so these two agree only for real-valued vectors.</p><p>When interacting with <code>Adjoint</code> vectors, this always obeys <code>(x ⊡ y)&#39; == y&#39; ⊡ x&#39;</code>, and hence may sometimes return another <code>Adjoint</code> vector. (And similarly for <code>Transpose</code>.)</p><pre><code class="language-julia-repl">julia&gt; M = rand(5,5); v = rand(5);

julia&gt; typeof(v ⊡ M&#39;)
Array{Float64,1}
Vector{Float64} (alias for Array{Float64, 1})

julia&gt; typeof(M ⊡ v&#39;) # adjoint of the previous line
Adjoint{Float64,Array{Float64,1}}
LinearAlgebra.Adjoint{Float64, Vector{Float64}}

julia&gt; typeof(v&#39; ⊡ M&#39;) # same as *, and equal to adjoint(M ⊡ v)
Adjoint{Float64,Array{Float64,1}}
LinearAlgebra.Adjoint{Float64, Vector{Float64}}

julia&gt; typeof(v&#39; ⊡ v)
Float64</code></pre><p>See also <code>boxdot!(Y,A,B)</code>, which is to <code>⊡</code> as <code>mul!</code> is to <code>*</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/TensorCore.jl/blob/7e9ad327756cfe3ded97a1c21c5e52c93a1d4db5/src/TensorCore.jl#LL134-L179">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="TensorCore.boxdot!" href="#TensorCore.boxdot!"><code>TensorCore.boxdot!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">boxdot!(Y, A, B, α=1, β=0)</code></pre><p>In-place version of <code>boxdot</code>, i.e. <code>Y .= (A ⊡ B) .* β .+ Y .* α</code>. Like 5-argument <code>mul!</code>, the use of <code>α, β</code> here requires Julia 1.3 or later.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/TensorCore.jl/blob/7e9ad327756cfe3ded97a1c21c5e52c93a1d4db5/src/TensorCore.jl#LL253-L258">source</a></section></article></article><nav class="docs-footer"><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></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Saturday 26 December 2020 20:48">Saturday 26 December 2020</span>. Using Julia version 1.5.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
Float64</code></pre><p>See also <code>boxdot!(Y,A,B)</code>, which is to <code>⊡</code> as <code>mul!</code> is to <code>*</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/TensorCore.jl/blob/83a840a962dcb64c64d9cb1346e82c83a4afaab1/src/TensorCore.jl#LL134-L179">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="TensorCore.boxdot!" href="#TensorCore.boxdot!"><code>TensorCore.boxdot!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia">boxdot!(Y, A, B, α=1, β=0)</code></pre><p>In-place version of <code>boxdot</code>, i.e. <code>Y .= (A ⊡ B) .* β .+ Y .* α</code>. Like 5-argument <code>mul!</code>, the use of <code>α, β</code> here requires Julia 1.3 or later.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaMath/TensorCore.jl/blob/83a840a962dcb64c64d9cb1346e82c83a4afaab1/src/TensorCore.jl#LL253-L258">source</a></section></article></article><nav class="docs-footer"><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></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Monday 12 February 2024 03:19">Monday 12 February 2024</span>. Using Julia version 1.10.0.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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