.. _sphx_glr_api_gallery_linalg_classical:

Classical matrix theory
-----------------------

Cramer's rule and the Cayley-Hamilton theorem: the determinant-era
results that predate numerical linear algebra.


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    <div class="sphx-glr-thumbcontainer" tooltip="Cramer&#x27;s rule gives every unknown as a ratio of two determinants, x_i = det(A_i) / det(A). It is exact in theory but needs n + 1 determinants, each an O(n^3) factorization, so its cost grows like n^4 against elimination&#x27;s n^3.">

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  .. image:: /api/gallery/linalg/classical/images/thumb/sphx_glr_plot_01_cramers_rule_thumb.png
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  :doc:`/api/gallery/linalg/classical/plot_01_cramers_rule`

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      <div class="sphx-glr-thumbnail-title">Cramer's rule vs. elimination</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Every square matrix satisfies its own characteristic equation: if p(l) = det(l I - A), then p(A) = 0. A consequence is that A^{-1} (and every higher power of A) is a polynomial in A of degree below n.">

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  .. image:: /api/gallery/linalg/classical/images/thumb/sphx_glr_plot_02_cayley_hamilton_thumb.png
    :alt:

  :doc:`/api/gallery/linalg/classical/plot_02_cayley_hamilton`

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      <div class="sphx-glr-thumbnail-title">The Cayley-Hamilton theorem</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/linalg/classical/plot_01_cramers_rule
   /api/gallery/linalg/classical/plot_02_cayley_hamilton