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

.. py:module:: toqito.perms.vec

.. autoapi-nested-parse::

   Performs the vec operation on a matrix.





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

.. py:function:: vec(mat)

   Perform the vec operation on a matrix.

   For more info, see Section: The Operator-Vector Correspondence from [@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:

   \[
       X =
       \begin{pmatrix}
           1 & 2 \\
           3 & 4
       \end{pmatrix}
   \]

   it holds that

   \[
       \text{vec}(X) = \begin{pmatrix} 1 & 3 & 2 & 4 \end{pmatrix}^T
   \]

   More formally, the vec operation is defined by

   \[
       \text{vec}(E_{a,b}) = e_a \otimes e_b
   \]

   for all \(a\) and \(b\) where

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

   for all \(c\) and \(d\) and where

   \[
       e_a(b) = \begin{cases}
                    1 & \text{if} \ a = b \\
                    0 & \text{if} \ a \not= b
                \end{cases}
   \]

   for all \(a\) and \(b\).

   .. rubric:: Examples

   Consider the following matrix

   \[
       A = \begin{pmatrix}
               1 & 2 \\
               3 & 4
           \end{pmatrix}
   \]

   Performing the \(\text{vec}\) operation on \(A\) yields

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

   ```python exec="1" source="above"
   import numpy as np
   from toqito.perms import vec

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

   print(vec(X))
   ```

   !!! See Also
       [`unvec()`][toqito.matrix_ops.unvec.unvec]

   :param mat: The input matrix.

   :returns: The vec representation of the matrix.