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Showing content from https://docs.scipy.org/doc/numpy-1.17.0/reference/generated/numpy.matmul.html below:

numpy.matmul — NumPy v1.17 Manual

Matrix product of two arrays.

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

y : ndarray

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

ValueError

If the last dimension of a is not the same size as the second-to-last dimension of b.

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 PEP465.

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 ...

New in version 1.10.0.

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