:py:mod:`matrix_props.is_commuting`
===================================

.. py:module:: matrix_props.is_commuting

.. autoapi-nested-parse::

   Is matrix commuting.



Module Contents
---------------


Functions
~~~~~~~~~

.. autoapisummary::

   matrix_props.is_commuting.is_commuting



.. py:function:: is_commuting(mat_1, mat_2)

   Determine if two linear operators commute with each other :cite:`WikiComm`.

   For any pair of operators :math:`X, Y \in \text{L}(\mathcal{X})`, the
   Lie bracket :math:`\left[X, Y\right] \in \text{L}(\mathcal{X})` is defined
   as

   .. math::
       \left[X, Y\right] = XY - YX.

   It holds that :math:`\left[X,Y\right]=0` if and only if :math:`X` and
   :math:`Y` commute (Section: Lie Brackets And Commutants from :cite:`Watrous_2018_TQI`).

   .. rubric:: Examples

   Consider the following matrices:

   .. math::
       A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},
       \quad \text{and} \quad
       B = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}.

   It holds that :math:`AB=0`, however

   .. math::
       BA = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = A,

   and hence, do not commute.

   >>> from toqito.matrix_props import is_commuting
   >>> import numpy as np
   >>> mat_1 = np.array([[0, 1], [0, 0]])
   >>> mat_2 = np.array([[1, 0], [0, 0]])
   >>> is_commuting(mat_1, mat_2)
   False

   Consider the following pair of matrices:

   .. math::
       A = \begin{pmatrix}
           1 & 0 & 0 \\
           0 & 1 & 0 \\
           1 & 0 & 2
           \end{pmatrix} \quad \text{and} \quad
       B = \begin{pmatrix}
           2 & 4 & 0 \\
           3 & 1 & 0 \\
           -1 & -4 & 1
           \end{pmatrix}.

   It may be verified that :math:`AB = BA = 0`, and therefore :math:`A` and
   :math:`B` commute.

   >>> from toqito.matrix_props import is_commuting
   >>> import numpy as np
   >>> mat_1 = np.array([[1, 0, 0], [0, 1, 0], [1, 0, 2]])
   >>> mat_2 = np.array([[2, 4, 0], [3, 1, 0], [-1, -4, 1]])
   >>> is_commuting(mat_1, mat_2)
   True

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :param mat_1: First matrix to check.
   :param mat_2: Second matrix to check.
   :return: Return `True` if :code:`mat_1` commutes with :code:`mat_2` and False otherwise.