# numpy.matmul

numpy.matmul(x1, x2, /, out=None, *, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj, axes, axis]) = <ufunc 'matmul'>

  Matrix product of two arrays.

  Parameters:
    x1, x2 array_like

      Input arrays, scalars not allowed.

    out ndarray, optional

      A location into which the result is stored. If provided, it must have a shape that matches the signature (n,k),(k,m)->(n,m). If not provided or None, a freshly-allocated array is returned.

    **kwargs

      For other keyword-only arguments, see the ufunc docs.

      New in version 1.16: Now handles ufunc kwargs

  Returns:
    y ndarray

      The matrix product of the inputs. This is a scalar only when both x1, x2 are 1-d vectors.

  Raises:
    ValueError

      If the last dimension of x1 is not the same size as the second-to-last dimension of x2.

      If a scalar value is passed in.

  See also

  `vdot`

    Complex-conjugating dot product.

  `tensordot`

    Sum products over arbitrary axes.

  `einsum`

    Einstein summation convention.

  `dot`

    alternative matrix product with different broadcasting rules.

  Notes

  The behavior depends on the arguments in the following way.

  - If both arguments are 2-D they are multiplied like conventional matrices.

  - If either argument is N-D, N > 2, it is treated as a stack of matrices residing in the last two indexes and broadcast accordingly.

  - If the first argument is 1-D, it is promoted to a matrix by prepending a 1 to its dimensions. After matrix multiplication the prepended 1 is removed.

  - If the second argument is 1-D, it is promoted to a matrix by appending a 1 to its dimensions. After matrix multiplication the appended 1 is removed.

  `matmul` differs from `dot` in two important ways:

  - Multiplication by scalars is not allowed, use `*` instead.

  - Stacks of matrices are broadcast together as if the matrices were elements, respecting the signature `(n,k),(k,m)->(n,m)`:

        >>> a = np.ones([9, 5, 7, 4])
        >>> c = np.ones([9, 5, 4, 3])
        >>> np.dot(a, c).shape
        (9, 5, 7, 9, 5, 3)
        >>> np.matmul(a, c).shape
        (9, 5, 7, 3)
        >>> # n is 7, k is 4, m is 3

  The matmul function implements the semantics of the `@` operator introduced in Python 3.5 following PEP 465.

  It uses an optimized BLAS library when possible (see `numpy.linalg`).

  Examples

  For 2-D arrays it is the matrix product:

      >>> a = np.array([[1, 0],
      ...               [0, 1]])
      >>> b = np.array([[4, 1],
      ...               [2, 2]])
      >>> np.matmul(a, b)
      array([[4, 1],
             [2, 2]])

  For 2-D mixed with 1-D, the result is the usual.

      >>> a = np.array([[1, 0],
      ...               [0, 1]])
      >>> b = np.array([1, 2])
      >>> np.matmul(a, b)
      array([1, 2])
      >>> np.matmul(b, a)
      array([1, 2])

  Broadcasting is conventional for stacks of arrays

      >>> a = np.arange(2 * 2 * 4).reshape((2, 2, 4))
      >>> b = np.arange(2 * 2 * 4).reshape((2, 4, 2))
      >>> np.matmul(a,b).shape
      (2, 2, 2)
      >>> np.matmul(a, b)[0, 1, 1]
      98
      >>> sum(a[0, 1, :] * b[0 , :, 1])
      98

  Vector, vector returns the scalar inner product, but neither argument is complex-conjugated:

      >>> np.matmul([2j, 3j], [2j, 3j])
      (-13+0j)

  Scalar multiplication raises an error.

      >>> np.matmul([1,2], 3)
      Traceback (most recent call last):
      ...
      ValueError: matmul: Input operand 1 does not have enough dimensions ...

  The `@` operator can be used as a shorthand for `np.matmul` on ndarrays.

      >>> x1 = np.array([2j, 3j])
      >>> x2 = np.array([2j, 3j])
      >>> x1 @ x2
      (-13+0j)

  New in version 1.10.0.