numpy.matmul#

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

Matrix product of two arrays.

Parameters:
x1, x2array_like

Input arrays, scalars not allowed.

outndarray, 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.

Returns:
yndarray

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

vecdot

Complex-conjugating dot product for stacks of vectors.

matvec

Matrix-vector product for stacks of matrices and vectors.

vecmat

Vector-matrix product for stacks of vectors and matrices.

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. (For stacks of vectors, use vecmat.)

  • 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. (For stacks of vectors, use matvec.)

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:

>>> import numpy as np

>>> 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)
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