# numpy.matmul¶

`numpy.``matmul`(x1, x2, /, out=None, *, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj]) = <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. ..versionadded:: 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.

`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.

numpy.outer

numpy.tensordot