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<section id="linear-algebra">
<span id="module-numpy.linalg"></span><span id="routines-linalg"></span><h1>Linear algebra<a class="headerlink" href="#linear-algebra" title="Link to this heading">#</a></h1>
<p>The NumPy linear algebra functions rely on BLAS and LAPACK to provide efficient
low level implementations of standard linear algebra algorithms. Those
libraries may be provided by NumPy itself using C versions of a subset of their
reference implementations but, when possible, highly optimized libraries that
take advantage of specialized processor functionality are preferred. Examples
of such libraries are <a class="reference external" href="https://www.openblas.net/">OpenBLAS</a>, MKL (TM), and ATLAS. Because those libraries
are multithreaded and processor dependent, environmental variables and external
packages such as <a class="reference external" href="https://github.com/joblib/threadpoolctl">threadpoolctl</a> may be needed to control the number of threads
or specify the processor architecture.</p>
<p>The SciPy library also contains a <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/linalg.html#module-scipy.linalg" title="(in SciPy v1.15.2)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg</span></code></a> submodule, and there is
overlap in the functionality provided by the SciPy and NumPy submodules. SciPy
contains functions not found in <a class="reference internal" href="#module-numpy.linalg" title="numpy.linalg"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.linalg</span></code></a>, such as functions related to
LU decomposition and the Schur decomposition, multiple ways of calculating the
pseudoinverse, and matrix transcendentals such as the matrix logarithm. Some
functions that exist in both have augmented functionality in <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/linalg.html#module-scipy.linalg" title="(in SciPy v1.15.2)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.linalg</span></code></a>.
For example, <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.eig.html#scipy.linalg.eig" title="(in SciPy v1.15.2)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.linalg.eig</span></code></a> can take a second matrix argument for solving
generalized eigenvalue problems. Some functions in NumPy, however, have more
flexible broadcasting options. For example, <a class="reference internal" href="generated/numpy.linalg.solve.html#numpy.linalg.solve" title="numpy.linalg.solve"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.linalg.solve</span></code></a> can handle
“stacked” arrays, while <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.solve.html#scipy.linalg.solve" title="(in SciPy v1.15.2)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.linalg.solve</span></code></a> accepts only a single square
array as its first argument.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The term <em>matrix</em> as it is used on this page indicates a 2d <a class="reference internal" href="generated/numpy.array.html#numpy.array" title="numpy.array"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.array</span></code></a>
object, and <em>not</em> a <a class="reference internal" href="generated/numpy.matrix.html#numpy.matrix" title="numpy.matrix"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.matrix</span></code></a> object. The latter is no longer
recommended, even for linear algebra. See
<a class="reference internal" href="arrays.classes.html#matrix-objects"><span class="std std-ref">the matrix object documentation</span></a> for
more information.</p>
</div>
<section id="the-operator">
<h2>The <code class="docutils literal notranslate"><span class="pre">@</span></code> operator<a class="headerlink" href="#the-operator" title="Link to this heading">#</a></h2>
<p>Introduced in NumPy 1.10.0, the <code class="docutils literal notranslate"><span class="pre">@</span></code> operator is preferable to
other methods when computing the matrix product between 2d arrays. The
<a class="reference internal" href="generated/numpy.matmul.html#numpy.matmul" title="numpy.matmul"><code class="xref py py-func docutils literal notranslate"><span class="pre">numpy.matmul</span></code></a> function implements the <code class="docutils literal notranslate"><span class="pre">@</span></code> operator.</p>
</section>
<section id="matrix-and-vector-products">
<h2>Matrix and vector products<a class="headerlink" href="#matrix-and-vector-products" title="Link to this heading">#</a></h2>
<div class="pst-scrollable-table-container"><table class="autosummary longtable table autosummary">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.dot.html#numpy.dot" title="numpy.dot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">dot</span></code></a>(a, b[, out])</p></td>
<td><p>Dot product of two arrays.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.multi_dot.html#numpy.linalg.multi_dot" title="numpy.linalg.multi_dot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.multi_dot</span></code></a>(arrays, *[, out])</p></td>
<td><p>Compute the dot product of two or more arrays in a single function call, while automatically selecting the fastest evaluation order.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.vdot.html#numpy.vdot" title="numpy.vdot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">vdot</span></code></a>(a, b, /)</p></td>
<td><p>Return the dot product of two vectors.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.vecdot.html#numpy.vecdot" title="numpy.vecdot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">vecdot</span></code></a>(x1, x2, /[, out, casting, order, ...])</p></td>
<td><p>Vector dot product of two arrays.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.vecdot.html#numpy.linalg.vecdot" title="numpy.linalg.vecdot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.vecdot</span></code></a>(x1, x2, /, *[, axis])</p></td>
<td><p>Computes the vector dot product.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.inner.html#numpy.inner" title="numpy.inner"><code class="xref py py-obj docutils literal notranslate"><span class="pre">inner</span></code></a>(a, b, /)</p></td>
<td><p>Inner product of two arrays.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.outer.html#numpy.outer" title="numpy.outer"><code class="xref py py-obj docutils literal notranslate"><span class="pre">outer</span></code></a>(a, b[, out])</p></td>
<td><p>Compute the outer product of two vectors.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.outer.html#numpy.linalg.outer" title="numpy.linalg.outer"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.outer</span></code></a>(x1, x2, /)</p></td>
<td><p>Compute the outer product of two vectors.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.matmul.html#numpy.matmul" title="numpy.matmul"><code class="xref py py-obj docutils literal notranslate"><span class="pre">matmul</span></code></a>(x1, x2, /[, out, casting, order, ...])</p></td>
<td><p>Matrix product of two arrays.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.matmul.html#numpy.linalg.matmul" title="numpy.linalg.matmul"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.matmul</span></code></a>(x1, x2, /)</p></td>
<td><p>Computes the matrix product.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.matvec.html#numpy.matvec" title="numpy.matvec"><code class="xref py py-obj docutils literal notranslate"><span class="pre">matvec</span></code></a>(x1, x2, /[, out, casting, order, ...])</p></td>
<td><p>Matrix-vector dot product of two arrays.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.vecmat.html#numpy.vecmat" title="numpy.vecmat"><code class="xref py py-obj docutils literal notranslate"><span class="pre">vecmat</span></code></a>(x1, x2, /[, out, casting, order, ...])</p></td>
<td><p>Vector-matrix dot product of two arrays.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.tensordot.html#numpy.tensordot" title="numpy.tensordot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">tensordot</span></code></a>(a, b[, axes])</p></td>
<td><p>Compute tensor dot product along specified axes.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.tensordot.html#numpy.linalg.tensordot" title="numpy.linalg.tensordot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.tensordot</span></code></a>(x1, x2, /, *[, axes])</p></td>
<td><p>Compute tensor dot product along specified axes.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.einsum.html#numpy.einsum" title="numpy.einsum"><code class="xref py py-obj docutils literal notranslate"><span class="pre">einsum</span></code></a>(subscripts, *operands[, out, dtype, ...])</p></td>
<td><p>Evaluates the Einstein summation convention on the operands.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.einsum_path.html#numpy.einsum_path" title="numpy.einsum_path"><code class="xref py py-obj docutils literal notranslate"><span class="pre">einsum_path</span></code></a>(subscripts, *operands[, optimize])</p></td>
<td><p>Evaluates the lowest cost contraction order for an einsum expression by considering the creation of intermediate arrays.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.matrix_power.html#numpy.linalg.matrix_power" title="numpy.linalg.matrix_power"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.matrix_power</span></code></a>(a, n)</p></td>
<td><p>Raise a square matrix to the (integer) power <em class="xref py py-obj">n</em>.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.kron.html#numpy.kron" title="numpy.kron"><code class="xref py py-obj docutils literal notranslate"><span class="pre">kron</span></code></a>(a, b)</p></td>
<td><p>Kronecker product of two arrays.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.cross.html#numpy.linalg.cross" title="numpy.linalg.cross"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.cross</span></code></a>(x1, x2, /, *[, axis])</p></td>
<td><p>Returns the cross product of 3-element vectors.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="decompositions">
<h2>Decompositions<a class="headerlink" href="#decompositions" title="Link to this heading">#</a></h2>
<div class="pst-scrollable-table-container"><table class="autosummary longtable table autosummary">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.cholesky.html#numpy.linalg.cholesky" title="numpy.linalg.cholesky"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.cholesky</span></code></a>(a, /, *[, upper])</p></td>
<td><p>Cholesky decomposition.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.qr.html#numpy.linalg.qr" title="numpy.linalg.qr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.qr</span></code></a>(a[, mode])</p></td>
<td><p>Compute the qr factorization of a matrix.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.svd.html#numpy.linalg.svd" title="numpy.linalg.svd"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.svd</span></code></a>(a[, full_matrices, compute_uv, ...])</p></td>
<td><p>Singular Value Decomposition.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.svdvals.html#numpy.linalg.svdvals" title="numpy.linalg.svdvals"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.svdvals</span></code></a>(x, /)</p></td>
<td><p>Returns the singular values of a matrix (or a stack of matrices) <code class="docutils literal notranslate"><span class="pre">x</span></code>.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="matrix-eigenvalues">
<h2>Matrix eigenvalues<a class="headerlink" href="#matrix-eigenvalues" title="Link to this heading">#</a></h2>
<div class="pst-scrollable-table-container"><table class="autosummary longtable table autosummary">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.eig.html#numpy.linalg.eig" title="numpy.linalg.eig"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.eig</span></code></a>(a)</p></td>
<td><p>Compute the eigenvalues and right eigenvectors of a square array.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.eigh.html#numpy.linalg.eigh" title="numpy.linalg.eigh"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.eigh</span></code></a>(a[, UPLO])</p></td>
<td><p>Return the eigenvalues and eigenvectors of a complex Hermitian (conjugate symmetric) or a real symmetric matrix.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.eigvals.html#numpy.linalg.eigvals" title="numpy.linalg.eigvals"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.eigvals</span></code></a>(a)</p></td>
<td><p>Compute the eigenvalues of a general matrix.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.eigvalsh.html#numpy.linalg.eigvalsh" title="numpy.linalg.eigvalsh"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.eigvalsh</span></code></a>(a[, UPLO])</p></td>
<td><p>Compute the eigenvalues of a complex Hermitian or real symmetric matrix.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="norms-and-other-numbers">
<h2>Norms and other numbers<a class="headerlink" href="#norms-and-other-numbers" title="Link to this heading">#</a></h2>
<div class="pst-scrollable-table-container"><table class="autosummary longtable table autosummary">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.norm.html#numpy.linalg.norm" title="numpy.linalg.norm"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.norm</span></code></a>(x[, ord, axis, keepdims])</p></td>
<td><p>Matrix or vector norm.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.matrix_norm.html#numpy.linalg.matrix_norm" title="numpy.linalg.matrix_norm"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.matrix_norm</span></code></a>(x, /, *[, keepdims, ord])</p></td>
<td><p>Computes the matrix norm of a matrix (or a stack of matrices) <code class="docutils literal notranslate"><span class="pre">x</span></code>.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.vector_norm.html#numpy.linalg.vector_norm" title="numpy.linalg.vector_norm"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.vector_norm</span></code></a>(x, /, *[, axis, ...])</p></td>
<td><p>Computes the vector norm of a vector (or batch of vectors) <code class="docutils literal notranslate"><span class="pre">x</span></code>.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.cond.html#numpy.linalg.cond" title="numpy.linalg.cond"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.cond</span></code></a>(x[, p])</p></td>
<td><p>Compute the condition number of a matrix.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.det.html#numpy.linalg.det" title="numpy.linalg.det"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.det</span></code></a>(a)</p></td>
<td><p>Compute the determinant of an array.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.matrix_rank.html#numpy.linalg.matrix_rank" title="numpy.linalg.matrix_rank"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.matrix_rank</span></code></a>(A[, tol, hermitian, rtol])</p></td>
<td><p>Return matrix rank of array using SVD method</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.slogdet.html#numpy.linalg.slogdet" title="numpy.linalg.slogdet"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.slogdet</span></code></a>(a)</p></td>
<td><p>Compute the sign and (natural) logarithm of the determinant of an array.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.trace.html#numpy.trace" title="numpy.trace"><code class="xref py py-obj docutils literal notranslate"><span class="pre">trace</span></code></a>(a[, offset, axis1, axis2, dtype, out])</p></td>
<td><p>Return the sum along diagonals of the array.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.trace.html#numpy.linalg.trace" title="numpy.linalg.trace"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.trace</span></code></a>(x, /, *[, offset, dtype])</p></td>
<td><p>Returns the sum along the specified diagonals of a matrix (or a stack of matrices) <code class="docutils literal notranslate"><span class="pre">x</span></code>.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="solving-equations-and-inverting-matrices">
<h2>Solving equations and inverting matrices<a class="headerlink" href="#solving-equations-and-inverting-matrices" title="Link to this heading">#</a></h2>
<div class="pst-scrollable-table-container"><table class="autosummary longtable table autosummary">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.solve.html#numpy.linalg.solve" title="numpy.linalg.solve"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.solve</span></code></a>(a, b)</p></td>
<td><p>Solve a linear matrix equation, or system of linear scalar equations.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.tensorsolve.html#numpy.linalg.tensorsolve" title="numpy.linalg.tensorsolve"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.tensorsolve</span></code></a>(a, b[, axes])</p></td>
<td><p>Solve the tensor equation <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">x</span> <span class="pre">=</span> <span class="pre">b</span></code> for x.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.lstsq.html#numpy.linalg.lstsq" title="numpy.linalg.lstsq"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.lstsq</span></code></a>(a, b[, rcond])</p></td>
<td><p>Return the least-squares solution to a linear matrix equation.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.inv.html#numpy.linalg.inv" title="numpy.linalg.inv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.inv</span></code></a>(a)</p></td>
<td><p>Compute the inverse of a matrix.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.pinv.html#numpy.linalg.pinv" title="numpy.linalg.pinv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.pinv</span></code></a>(a[, rcond, hermitian, rtol])</p></td>
<td><p>Compute the (Moore-Penrose) pseudo-inverse of a matrix.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.tensorinv.html#numpy.linalg.tensorinv" title="numpy.linalg.tensorinv"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.tensorinv</span></code></a>(a[, ind])</p></td>
<td><p>Compute the 'inverse' of an N-dimensional array.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="other-matrix-operations">
<h2>Other matrix operations<a class="headerlink" href="#other-matrix-operations" title="Link to this heading">#</a></h2>
<div class="pst-scrollable-table-container"><table class="autosummary longtable table autosummary">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.diagonal.html#numpy.diagonal" title="numpy.diagonal"><code class="xref py py-obj docutils literal notranslate"><span class="pre">diagonal</span></code></a>(a[, offset, axis1, axis2])</p></td>
<td><p>Return specified diagonals.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.linalg.diagonal.html#numpy.linalg.diagonal" title="numpy.linalg.diagonal"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.diagonal</span></code></a>(x, /, *[, offset])</p></td>
<td><p>Returns specified diagonals of a matrix (or a stack of matrices) <code class="docutils literal notranslate"><span class="pre">x</span></code>.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.matrix_transpose.html#numpy.linalg.matrix_transpose" title="numpy.linalg.matrix_transpose"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.matrix_transpose</span></code></a>(x, /)</p></td>
<td><p>Transposes a matrix (or a stack of matrices) <code class="docutils literal notranslate"><span class="pre">x</span></code>.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="exceptions">
<h2>Exceptions<a class="headerlink" href="#exceptions" title="Link to this heading">#</a></h2>
<div class="pst-scrollable-table-container"><table class="autosummary longtable table autosummary">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.linalg.LinAlgError.html#numpy.linalg.LinAlgError" title="numpy.linalg.LinAlgError"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.LinAlgError</span></code></a></p></td>
<td><p>Generic Python-exception-derived object raised by linalg functions.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="linear-algebra-on-several-matrices-at-once">
<span id="routines-linalg-broadcasting"></span><h2>Linear algebra on several matrices at once<a class="headerlink" href="#linear-algebra-on-several-matrices-at-once" title="Link to this heading">#</a></h2>
<p>Several of the linear algebra routines listed above are able to
compute results for several matrices at once, if they are stacked into
the same array.</p>
<p>This is indicated in the documentation via input parameter
specifications such as <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">:</span> <span class="pre">(...,</span> <span class="pre">M,</span> <span class="pre">M)</span> <span class="pre">array_like</span></code>. This means that
if for instance given an input array <code class="docutils literal notranslate"><span class="pre">a.shape</span> <span class="pre">==</span> <span class="pre">(N,</span> <span class="pre">M,</span> <span class="pre">M)</span></code>, it is
interpreted as a “stack” of N matrices, each of size M-by-M. Similar
specification applies to return values, for instance the determinant
has <code class="docutils literal notranslate"><span class="pre">det</span> <span class="pre">:</span> <span class="pre">(...)</span></code> and will in this case return an array of shape
<code class="docutils literal notranslate"><span class="pre">det(a).shape</span> <span class="pre">==</span> <span class="pre">(N,)</span></code>. This generalizes to linear algebra
operations on higher-dimensional arrays: the last 1 or 2 dimensions of
a multidimensional array are interpreted as vectors or matrices, as
appropriate for each operation.</p>
</section>
</section>
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