:py:mod:`matrix_ops.inner_product`
==================================

.. py:module:: matrix_ops.inner_product

.. autoapi-nested-parse::

   Inner product operation.



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


Functions
~~~~~~~~~

.. autoapisummary::

   matrix_ops.inner_product.inner_product



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

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

   The inner product is calculated as follows:

   .. math::
       \left\langle
       \begin{pmatrix}
           a_1 \\ \vdots \\ a_n
       \end{pmatrix},
       \begin{pmatrix}
           b_1 \\ \vdots \\ b_n
       \end{pmatrix} \right\rangle =
       \begin{pmatrix}
           a_1, \cdots, a_n
       \end{pmatrix}
       \begin{pmatrix}
           b_1 \\ \vdots \\ b_n
       \end{pmatrix} = a_1 b_1 + \cdots + a_n b_n

   .. rubric:: Example

   The inner 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\langle
       \begin{pmatrix}
           1 \\ 2 \\ 3
       \end{pmatrix},
       \begin{pmatrix}
           4 \\ 5 \\ 6
       \end{pmatrix}\right\rangle =
       \begin{pmatrix}
           1, 2, 3
       \end{pmatrix}
       \begin{pmatrix}
           4 \\ 5 \\ 6
       \end{pmatrix} = 1\times 4 + 2\times 5 + 3\times 6 =
       32

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

   >>> import numpy as np
   >>> from toqito.matrix_ops import inner_product
   >>> v1, v2 = np.array([1,2,3]), np.array([4,5,6])
   >>> inner_product(v1,v2)
   32

   .. 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 inner product.