perms.vec
=========

.. py:module:: perms.vec

.. autoapi-nested-parse::

   Performs the vec operation on a matrix.



Functions
---------

.. autoapisummary::

   perms.vec.vec


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

.. py:function:: vec(mat)

   Perform the vec operation on a matrix.

   For more info, see Section: The Operator-Vector Correspondence from :footcite:`Watrous_2018_TQI`.

   The function reorders the given matrix into a column vector by stacking the columns of the matrix sequentially.

   For instance, for the following matrix:

   .. math::
       X =
       \begin{pmatrix}
           1 & 2 \\
           3 & 4
       \end{pmatrix}

   it holds that

   .. math::
       \text{vec}(X) = \begin{pmatrix} 1 & 3 & 2 & 4 \end{pmatrix}^T

   More formally, the vec operation is defined by

   .. math::
       \text{vec}(E_{a,b}) = e_a \otimes e_b

   for all :math:`a` and :math:`b` where

   .. math::
       E_{a,b}(c,d) = \begin{cases}
                         1 & \text{if} \ (c,d) = (a,b) \\
                         0 & \text{otherwise}
                       \end{cases}

   for all :math:`c` and :math:`d` and where

   .. math::
       e_a(b) = \begin{cases}
                    1 & \text{if} \ a = b \\
                    0 & \text{if} \ a \not= b
                \end{cases}

   for all :math:`a` and :math:`b`.

   .. rubric:: Examples

   Consider the following matrix

   .. math::
       A = \begin{pmatrix}
               1 & 2 \\
               3 & 4
           \end{pmatrix}

   Performing the :math:`\text{vec}` operation on :math:`A` yields

   .. math::
       \text{vec}(A) = \left[1, 3, 2, 4 \right]^{T}.

   .. jupyter-execute::

    import numpy as np
    from toqito.perms import vec

    X = np.array([[1, 2], [3, 4]])

    vec(X)

   .. seealso:: :func:`.unvec`

   .. rubric:: References

   .. footbibliography::

   :param mat: The input matrix.
   :return: The vec representation of the matrix.