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<section id="torch-linalg">
<h1>torch.linalg<a class="headerlink" href="#torch-linalg" title="Permalink to this heading">¶</a></h1>
<p>Common linear algebra operations.</p>
<p>See <a class="reference internal" href="notes/numerical_accuracy.html#linear-algebra-stability"><span class="std std-ref">Linear algebra (torch.linalg)</span></a> for some common numerical edge-cases.</p>
<span class="target" id="module-torch.linalg"></span><section id="matrix-properties">
<h2>Matrix Properties<a class="headerlink" href="#matrix-properties" title="Permalink to this heading">¶</a></h2>
<table class="autosummary longtable docutils align-default">
<tbody>
<tr class="row-odd"><td><p><p id="torch.linalg.norm"/><a class="reference internal" href="generated/torch.linalg.norm.html#torch.linalg.norm" title="torch.linalg.norm"><code class="xref py py-obj docutils literal notranslate"><span class="pre">norm</span></code></a></p></td>
<td><p>Computes a vector or matrix norm.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.vector_norm"/><a class="reference internal" href="generated/torch.linalg.vector_norm.html#torch.linalg.vector_norm" title="torch.linalg.vector_norm"><code class="xref py py-obj docutils literal notranslate"><span class="pre">vector_norm</span></code></a></p></td>
<td><p>Computes a vector norm.</p></td>
</tr>
<tr class="row-odd"><td><p><p id="torch.linalg.matrix_norm"/><a class="reference internal" href="generated/torch.linalg.matrix_norm.html#torch.linalg.matrix_norm" title="torch.linalg.matrix_norm"><code class="xref py py-obj docutils literal notranslate"><span class="pre">matrix_norm</span></code></a></p></td>
<td><p>Computes a matrix norm.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.diagonal"/><a class="reference internal" href="generated/torch.linalg.diagonal.html#torch.linalg.diagonal" title="torch.linalg.diagonal"><code class="xref py py-obj docutils literal notranslate"><span class="pre">diagonal</span></code></a></p></td>
<td><p>Alias for <a class="reference internal" href="generated/torch.diagonal.html#torch.diagonal" title="torch.diagonal"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.diagonal()</span></code></a> with defaults <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim1</span></code><cite>= -2</cite>, <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim2</span></code><cite>= -1</cite>.</p></td>
</tr>
<tr class="row-odd"><td><p><p id="torch.linalg.det"/><a class="reference internal" href="generated/torch.linalg.det.html#torch.linalg.det" title="torch.linalg.det"><code class="xref py py-obj docutils literal notranslate"><span class="pre">det</span></code></a></p></td>
<td><p>Computes the determinant of a square matrix.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.slogdet"/><a class="reference internal" href="generated/torch.linalg.slogdet.html#torch.linalg.slogdet" title="torch.linalg.slogdet"><code class="xref py py-obj docutils literal notranslate"><span class="pre">slogdet</span></code></a></p></td>
<td><p>Computes the sign and natural logarithm of the absolute value of the determinant of a square matrix.</p></td>
</tr>
<tr class="row-odd"><td><p><p id="torch.linalg.cond"/><a class="reference internal" href="generated/torch.linalg.cond.html#torch.linalg.cond" title="torch.linalg.cond"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cond</span></code></a></p></td>
<td><p>Computes the condition number of a matrix with respect to a matrix norm.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.matrix_rank"/><a class="reference internal" href="generated/torch.linalg.matrix_rank.html#torch.linalg.matrix_rank" title="torch.linalg.matrix_rank"><code class="xref py py-obj docutils literal notranslate"><span class="pre">matrix_rank</span></code></a></p></td>
<td><p>Computes the numerical rank of a matrix.</p></td>
</tr>
</tbody>
</table>
</section>
<section id="decompositions">
<h2>Decompositions<a class="headerlink" href="#decompositions" title="Permalink to this heading">¶</a></h2>
<table class="autosummary longtable docutils align-default">
<tbody>
<tr class="row-odd"><td><p><p id="torch.linalg.cholesky"/><a class="reference internal" href="generated/torch.linalg.cholesky.html#torch.linalg.cholesky" title="torch.linalg.cholesky"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cholesky</span></code></a></p></td>
<td><p>Computes the Cholesky decomposition of a complex Hermitian or real symmetric positive-definite matrix.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.qr"/><a class="reference internal" href="generated/torch.linalg.qr.html#torch.linalg.qr" title="torch.linalg.qr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">qr</span></code></a></p></td>
<td><p>Computes the QR decomposition of a matrix.</p></td>
</tr>
<tr class="row-odd"><td><p><p id="torch.linalg.lu"/><a class="reference internal" href="generated/torch.linalg.lu.html#torch.linalg.lu" title="torch.linalg.lu"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lu</span></code></a></p></td>
<td><p>Computes the LU decomposition with partial pivoting of a matrix.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.lu_factor"/><a class="reference internal" href="generated/torch.linalg.lu_factor.html#torch.linalg.lu_factor" title="torch.linalg.lu_factor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lu_factor</span></code></a></p></td>
<td><p>Computes a compact representation of the LU factorization with partial pivoting of a matrix.</p></td>
</tr>
<tr class="row-odd"><td><p><p id="torch.linalg.eig"/><a class="reference internal" href="generated/torch.linalg.eig.html#torch.linalg.eig" title="torch.linalg.eig"><code class="xref py py-obj docutils literal notranslate"><span class="pre">eig</span></code></a></p></td>
<td><p>Computes the eigenvalue decomposition of a square matrix if it exists.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.eigvals"/><a class="reference internal" href="generated/torch.linalg.eigvals.html#torch.linalg.eigvals" title="torch.linalg.eigvals"><code class="xref py py-obj docutils literal notranslate"><span class="pre">eigvals</span></code></a></p></td>
<td><p>Computes the eigenvalues of a square matrix.</p></td>
</tr>
<tr class="row-odd"><td><p><p id="torch.linalg.eigh"/><a class="reference internal" href="generated/torch.linalg.eigh.html#torch.linalg.eigh" title="torch.linalg.eigh"><code class="xref py py-obj docutils literal notranslate"><span class="pre">eigh</span></code></a></p></td>
<td><p>Computes the eigenvalue decomposition of a complex Hermitian or real symmetric matrix.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.eigvalsh"/><a class="reference internal" href="generated/torch.linalg.eigvalsh.html#torch.linalg.eigvalsh" title="torch.linalg.eigvalsh"><code class="xref py py-obj docutils literal notranslate"><span class="pre">eigvalsh</span></code></a></p></td>
<td><p>Computes the eigenvalues of a complex Hermitian or real symmetric matrix.</p></td>
</tr>
<tr class="row-odd"><td><p><p id="torch.linalg.svd"/><a class="reference internal" href="generated/torch.linalg.svd.html#torch.linalg.svd" title="torch.linalg.svd"><code class="xref py py-obj docutils literal notranslate"><span class="pre">svd</span></code></a></p></td>
<td><p>Computes the singular value decomposition (SVD) of a matrix.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.svdvals"/><a class="reference internal" href="generated/torch.linalg.svdvals.html#torch.linalg.svdvals" title="torch.linalg.svdvals"><code class="xref py py-obj docutils literal notranslate"><span class="pre">svdvals</span></code></a></p></td>
<td><p>Computes the singular values of a matrix.</p></td>
</tr>
</tbody>
</table>
</section>
<section id="solvers">
<span id="linalg-solvers"></span><h2>Solvers<a class="headerlink" href="#solvers" title="Permalink to this heading">¶</a></h2>
<table class="autosummary longtable docutils align-default">
<tbody>
<tr class="row-odd"><td><p><p id="torch.linalg.solve"/><a class="reference internal" href="generated/torch.linalg.solve.html#torch.linalg.solve" title="torch.linalg.solve"><code class="xref py py-obj docutils literal notranslate"><span class="pre">solve</span></code></a></p></td>
<td><p>Computes the solution of a square system of linear equations with a unique solution.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.solve_triangular"/><a class="reference internal" href="generated/torch.linalg.solve_triangular.html#torch.linalg.solve_triangular" title="torch.linalg.solve_triangular"><code class="xref py py-obj docutils literal notranslate"><span class="pre">solve_triangular</span></code></a></p></td>
<td><p>Computes the solution of a triangular system of linear equations with a unique solution.</p></td>
</tr>
<tr class="row-odd"><td><p><p id="torch.linalg.lu_solve"/><a class="reference internal" href="generated/torch.linalg.lu_solve.html#torch.linalg.lu_solve" title="torch.linalg.lu_solve"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lu_solve</span></code></a></p></td>
<td><p>Computes the solution of a square system of linear equations with a unique solution given an LU decomposition.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.lstsq"/><a class="reference internal" href="generated/torch.linalg.lstsq.html#torch.linalg.lstsq" title="torch.linalg.lstsq"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lstsq</span></code></a></p></td>
<td><p>Computes a solution to the least squares problem of a system of linear equations.</p></td>
</tr>
</tbody>
</table>
</section>
<section id="inverses">
<span id="linalg-inverses"></span><h2>Inverses<a class="headerlink" href="#inverses" title="Permalink to this heading">¶</a></h2>
<table class="autosummary longtable docutils align-default">
<tbody>
<tr class="row-odd"><td><p><p id="torch.linalg.inv"/><a class="reference internal" href="generated/torch.linalg.inv.html#torch.linalg.inv" title="torch.linalg.inv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">inv</span></code></a></p></td>
<td><p>Computes the inverse of a square matrix if it exists.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.pinv"/><a class="reference internal" href="generated/torch.linalg.pinv.html#torch.linalg.pinv" title="torch.linalg.pinv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">pinv</span></code></a></p></td>
<td><p>Computes the pseudoinverse (Moore-Penrose inverse) of a matrix.</p></td>
</tr>
</tbody>
</table>
</section>
<section id="matrix-functions">
<h2>Matrix Functions<a class="headerlink" href="#matrix-functions" title="Permalink to this heading">¶</a></h2>
<table class="autosummary longtable docutils align-default">
<tbody>
<tr class="row-odd"><td><p><p id="torch.linalg.matrix_exp"/><a class="reference internal" href="generated/torch.linalg.matrix_exp.html#torch.linalg.matrix_exp" title="torch.linalg.matrix_exp"><code class="xref py py-obj docutils literal notranslate"><span class="pre">matrix_exp</span></code></a></p></td>
<td><p>Computes the matrix exponential of a square matrix.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.matrix_power"/><a class="reference internal" href="generated/torch.linalg.matrix_power.html#torch.linalg.matrix_power" title="torch.linalg.matrix_power"><code class="xref py py-obj docutils literal notranslate"><span class="pre">matrix_power</span></code></a></p></td>
<td><p>Computes the <cite>n</cite>-th power of a square matrix for an integer <cite>n</cite>.</p></td>
</tr>
</tbody>
</table>
</section>
<section id="matrix-products">
<h2>Matrix Products<a class="headerlink" href="#matrix-products" title="Permalink to this heading">¶</a></h2>
<table class="autosummary longtable docutils align-default">
<tbody>
<tr class="row-odd"><td><p><p id="torch.linalg.cross"/><a class="reference internal" href="generated/torch.linalg.cross.html#torch.linalg.cross" title="torch.linalg.cross"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cross</span></code></a></p></td>
<td><p>Computes the cross product of two 3-dimensional vectors.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.matmul"/><a class="reference internal" href="generated/torch.linalg.matmul.html#torch.linalg.matmul" title="torch.linalg.matmul"><code class="xref py py-obj docutils literal notranslate"><span class="pre">matmul</span></code></a></p></td>
<td><p>Alias for <a class="reference internal" href="generated/torch.matmul.html#torch.matmul" title="torch.matmul"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.matmul()</span></code></a></p></td>
</tr>
<tr class="row-odd"><td><p><p id="torch.linalg.vecdot"/><a class="reference internal" href="generated/torch.linalg.vecdot.html#torch.linalg.vecdot" title="torch.linalg.vecdot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">vecdot</span></code></a></p></td>
<td><p>Computes the dot product of two batches of vectors along a dimension.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.multi_dot"/><a class="reference internal" href="generated/torch.linalg.multi_dot.html#torch.linalg.multi_dot" title="torch.linalg.multi_dot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">multi_dot</span></code></a></p></td>
<td><p>Efficiently multiplies two or more matrices by reordering the multiplications so that the fewest arithmetic operations are performed.</p></td>
</tr>
<tr class="row-odd"><td><p><p id="torch.linalg.householder_product"/><a class="reference internal" href="generated/torch.linalg.householder_product.html#torch.linalg.householder_product" title="torch.linalg.householder_product"><code class="xref py py-obj docutils literal notranslate"><span class="pre">householder_product</span></code></a></p></td>
<td><p>Computes the first <cite>n</cite> columns of a product of Householder matrices.</p></td>
</tr>
</tbody>
</table>
</section>
<section id="tensor-operations">
<h2>Tensor Operations<a class="headerlink" href="#tensor-operations" title="Permalink to this heading">¶</a></h2>
<table class="autosummary longtable docutils align-default">
<tbody>
<tr class="row-odd"><td><p><p id="torch.linalg.tensorinv"/><a class="reference internal" href="generated/torch.linalg.tensorinv.html#torch.linalg.tensorinv" title="torch.linalg.tensorinv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">tensorinv</span></code></a></p></td>
<td><p>Computes the multiplicative inverse of <a class="reference internal" href="generated/torch.tensordot.html#torch.tensordot" title="torch.tensordot"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.tensordot()</span></code></a>.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.tensorsolve"/><a class="reference internal" href="generated/torch.linalg.tensorsolve.html#torch.linalg.tensorsolve" title="torch.linalg.tensorsolve"><code class="xref py py-obj docutils literal notranslate"><span class="pre">tensorsolve</span></code></a></p></td>
<td><p>Computes the solution <cite>X</cite> to the system <cite>torch.tensordot(A, X) = B</cite>.</p></td>
</tr>
</tbody>
</table>
</section>
<section id="misc">
<h2>Misc<a class="headerlink" href="#misc" title="Permalink to this heading">¶</a></h2>
<table class="autosummary longtable docutils align-default">
<tbody>
<tr class="row-odd"><td><p><p id="torch.linalg.vander"/><a class="reference internal" href="generated/torch.linalg.vander.html#torch.linalg.vander" title="torch.linalg.vander"><code class="xref py py-obj docutils literal notranslate"><span class="pre">vander</span></code></a></p></td>
<td><p>Generates a Vandermonde matrix.</p></td>
</tr>
</tbody>
</table>
</section>
<section id="experimental-functions">
<h2>Experimental Functions<a class="headerlink" href="#experimental-functions" title="Permalink to this heading">¶</a></h2>
<table class="autosummary longtable docutils align-default">
<tbody>
<tr class="row-odd"><td><p><p id="torch.linalg.cholesky_ex"/><a class="reference internal" href="generated/torch.linalg.cholesky_ex.html#torch.linalg.cholesky_ex" title="torch.linalg.cholesky_ex"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cholesky_ex</span></code></a></p></td>
<td><p>Computes the Cholesky decomposition of a complex Hermitian or real symmetric positive-definite matrix.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.inv_ex"/><a class="reference internal" href="generated/torch.linalg.inv_ex.html#torch.linalg.inv_ex" title="torch.linalg.inv_ex"><code class="xref py py-obj docutils literal notranslate"><span class="pre">inv_ex</span></code></a></p></td>
<td><p>Computes the inverse of a square matrix if it is invertible.</p></td>
</tr>
<tr class="row-odd"><td><p><p id="torch.linalg.solve_ex"/><a class="reference internal" href="generated/torch.linalg.solve_ex.html#torch.linalg.solve_ex" title="torch.linalg.solve_ex"><code class="xref py py-obj docutils literal notranslate"><span class="pre">solve_ex</span></code></a></p></td>
<td><p>A version of <a class="reference internal" href="generated/torch.linalg.solve.html#torch.linalg.solve" title="torch.linalg.solve"><code class="xref py py-func docutils literal notranslate"><span class="pre">solve()</span></code></a> that does not perform error checks unless <code class="xref py py-attr docutils literal notranslate"><span class="pre">check_errors</span></code><cite>= True</cite>.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.lu_factor_ex"/><a class="reference internal" href="generated/torch.linalg.lu_factor_ex.html#torch.linalg.lu_factor_ex" title="torch.linalg.lu_factor_ex"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lu_factor_ex</span></code></a></p></td>
<td><p>This is a version of <a class="reference internal" href="generated/torch.linalg.lu_factor.html#torch.linalg.lu_factor" title="torch.linalg.lu_factor"><code class="xref py py-func docutils literal notranslate"><span class="pre">lu_factor()</span></code></a> that does not perform error checks unless <code class="xref py py-attr docutils literal notranslate"><span class="pre">check_errors</span></code><cite>= True</cite>.</p></td>
</tr>
<tr class="row-odd"><td><p><p id="torch.linalg.ldl_factor"/><a class="reference internal" href="generated/torch.linalg.ldl_factor.html#torch.linalg.ldl_factor" title="torch.linalg.ldl_factor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ldl_factor</span></code></a></p></td>
<td><p>Computes a compact representation of the LDL factorization of a Hermitian or symmetric (possibly indefinite) matrix.</p></td>
</tr>
<tr class="row-even"><td><p><p id="torch.linalg.ldl_factor_ex"/><a class="reference internal" href="generated/torch.linalg.ldl_factor_ex.html#torch.linalg.ldl_factor_ex" title="torch.linalg.ldl_factor_ex"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ldl_factor_ex</span></code></a></p></td>
<td><p>This is a version of <a class="reference internal" href="generated/torch.linalg.ldl_factor.html#torch.linalg.ldl_factor" title="torch.linalg.ldl_factor"><code class="xref py py-func docutils literal notranslate"><span class="pre">ldl_factor()</span></code></a> that does not perform error checks unless <code class="xref py py-attr docutils literal notranslate"><span class="pre">check_errors</span></code><cite>= True</cite>.</p></td>
</tr>
<tr class="row-odd"><td><p><p id="torch.linalg.ldl_solve"/><a class="reference internal" href="generated/torch.linalg.ldl_solve.html#torch.linalg.ldl_solve" title="torch.linalg.ldl_solve"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ldl_solve</span></code></a></p></td>
<td><p>Computes the solution of a system of linear equations using the LDL factorization.</p></td>
</tr>
</tbody>
</table>
</section>
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