.. _sphx_glr_api_gallery_linalg_svd:

Singular value decomposition
-----------------------------

SVD via :func:`numpy.linalg.svd` and the Eckart-Young best low-rank approximation.

Also the Moore-Penrose pseudoinverse and minimum-norm least squares.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Truncating the SVD A = U diag(S) V^T to its k largest singular values gives the best rank-``k`` approximation of A: its error is sqrt(sigma_{k+1}^2 + ...) in the Frobenius norm and sigma_{k+1} in the spectral norm, and no other rank-``k`` matrix does better. The SVD itself comes from numpy.linalg.svd (LAPACK&#x27;s stable bidiagonalization route, not the eigendecomposition of A^T A).">

.. only:: html

  .. image:: /api/gallery/linalg/svd/images/thumb/sphx_glr_plot_01_svd_thumb.png
    :alt:

  :doc:`/api/gallery/linalg/svd/plot_01_svd`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Eckart-Young: the best low-rank approximation from the SVD</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="For a rectangular or rank-deficient A there is no inverse, but there is exactly one matrix A^+ satisfying Penrose&#x27;s four equations, and x = A^+ b is the shortest vector among all least-squares solutions.">

.. only:: html

  .. image:: /api/gallery/linalg/svd/images/thumb/sphx_glr_plot_02_moore_penrose_pseudoinverse_thumb.png
    :alt:

  :doc:`/api/gallery/linalg/svd/plot_02_moore_penrose_pseudoinverse`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Moore-Penrose pseudoinverse</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/linalg/svd/plot_01_svd
   /api/gallery/linalg/svd/plot_02_moore_penrose_pseudoinverse