:py:mod:`matrix_ops.outer_product`
==================================

.. py:module:: matrix_ops.outer_product

.. autoapi-nested-parse::

   Outer product operation.



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


Functions
~~~~~~~~~

.. autoapisummary::

   matrix_ops.outer_product.outer_product



.. py:function:: outer_product(v1, v2)

   Compute the outer product :math:`|v_1\rangle\langle v_2|` of two vectors.

   The outer product is calculated as follows :cite:`WikiOuterProd` :

   .. math::
       \left|\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}\right\rangle\left\langle\begin{pmatrix}b_1\\\vdots\\b_n\end{pmatrix}\right|=\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}\begin{pmatrix}b_1&\cdots&b_n\end{pmatrix}=\begin{pmatrix}a_1b_1&\cdots&a_1b_n\\\vdots&\ddots&\vdots\\a_1b_n&\cdots&a_nb_n\end{pmatrix}

   .. rubric:: Example

   The outer product of the vectors :math:`v1 = \begin{pmatrix}1 \\ 2 \\ 3 \end{pmatrix}` and :math:`v2 =
   \begin{pmatrix}4 \\ 5 \\ 6 \ \end{pmatrix}` looks as follows:

   .. math::
       \left|\begin{pmatrix}
           1 \\ 2 \\ 3
       \end{pmatrix}\right\rangle
       \left\langle
       \begin{pmatrix}
           4 \\ 5 \\ 6
       \end{pmatrix}\right|=
       \begin{pmatrix}
           1 \\ 2 \\ 3
       \end{pmatrix}
       \begin{pmatrix}
           4 & 5 & 6
       \end{pmatrix}=
       \begin{pmatrix}
           1 \times 4 & 1 \times 5 & 1 \times 6 \\
           2 \times 4 & 2 \times 5 & 2 \times 6 \\
           3 \times 4 & 3 \times 5 & 3 \times 6
       \end{pmatrix}=
       \begin{pmatrix}
           4 & 5 & 6 \\
           8 & 10 & 12 \\
           12 & 15 & 18
       \end{pmatrix}

   In :code:`toqito`, this looks like this:

   >>> import numpy as np
   >>> from toqito.matrix_ops import outer_product
   >>> v1, v2 = np.array([1,2,3]), np.array([4,5,6])
   >>> outer_product(v1,v2)
   array([[ 4,  5,  6],
          [ 8, 10, 12],
          [12, 15, 18]])

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :raises ValueError: Vector dimensions are mismatched.
   :param v1: v1 and v2, both vectors of dimensions :math:`(n,1)` where :math:`n>1`.
   :param v2: v1 and v2, both vectors of dimensions :math:`(n,1)` where :math:`n>1`.
   :return: The computed outer product.