.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/linalg/classical/plot_02_cayley_hamilton.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_linalg_classical_plot_02_cayley_hamilton.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_linalg_classical_plot_02_cayley_hamilton.py:


The Cayley-Hamilton theorem
================================

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``.

.. GENERATED FROM PYTHON SOURCE LINES 12-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.linalg import characteristic_polynomial, matrix_polynomial
    from mathematicskit.linalg.visualizers.plots import plot_matrix_heatmap








.. GENERATED FROM PYTHON SOURCE LINES 19-22

Cayley's 2x2 case
-----------------------
For a 2x2 matrix, p(l) = l^2 - tr(A) l + det(A).

.. GENERATED FROM PYTHON SOURCE LINES 22-28

.. code-block:: Python


    A = np.array([[1.0, 2.0], [3.0, 4.0]])
    p = characteristic_polynomial(A)
    print("p(l) coefficients:", np.round(p, 10))
    print("p(A) =\n", np.round(matrix_polynomial(p, A), 12))





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    p(l) coefficients: [ 1. -5. -2.]
    p(A) =
     [[0. 0.]
     [0. 0.]]




.. GENERATED FROM PYTHON SOURCE LINES 29-32

A 5x5 random matrix, and the inverse as a polynomial
---------------------------------------------------------
From p(A) = 0: A^{-1} = -(A^{n-1} + c_1 A^{n-2} + ... + c_{n-1} I) / c_n.

.. GENERATED FROM PYTHON SOURCE LINES 32-47

.. code-block:: Python


    rng = np.random.default_rng(1)
    B = rng.normal(size=(5, 5))
    c = characteristic_polynomial(B)
    residual = matrix_polynomial(c, B)
    print(f"||p(B)||_F = {np.linalg.norm(residual):.2e}")
    B_inv = -matrix_polynomial(c[:-1], B) / c[-1]
    print(f"||B^-1 (Cayley-Hamilton) - inv(B)||_F = {np.linalg.norm(B_inv - np.linalg.inv(B)):.2e}")

    fig, axes = plt.subplots(1, 2, figsize=(8, 3.5))
    plot_matrix_heatmap(B @ B, ax=axes[0], title="B^2")
    plot_matrix_heatmap(residual, ax=axes[1], title="p(B) (rounding error only)")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/linalg/classical/images/sphx_glr_plot_02_cayley_hamilton_001.png
   :alt: B^2, p(B) (rounding error only)
   :srcset: /api/gallery/linalg/classical/images/sphx_glr_plot_02_cayley_hamilton_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    ||p(B)||_F = 2.00e-14
    ||B^-1 (Cayley-Hamilton) - inv(B)||_F = 1.72e-14





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.044 seconds)


.. _sphx_glr_download_api_gallery_linalg_classical_plot_02_cayley_hamilton.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_cayley_hamilton.ipynb <plot_02_cayley_hamilton.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_cayley_hamilton.py <plot_02_cayley_hamilton.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_cayley_hamilton.zip <plot_02_cayley_hamilton.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_