einsum(subscripts, *operands, out=None, optimize=False)

Evaluates the Einstein summation convention on the operands.

Using the Einstein summation convention, many common multi-dimensional, linear algebraic array operations can be represented in a simple fashion. In implicit mode einsum computes these values.

In explicit mode, einsum provides further flexibility to compute other array operations that might not be considered classical Einstein summation operations, by disabling, or forcing summation over specified subscript labels.

See the notes and examples for clarification.

  • subscripts (str) – Specifies the subscripts for summation as comma separated list of subscript labels. An implicit (classical Einstein summation) calculation is performed unless the explicit indicator ‘->’ is included as well as subscript labels of the precise output form.

  • operands (list of ndarray) – These are the arrays for the operation.

  • out (ndarray, optional) – If provided, the calculation is done into this array.

  • optimize ({False, True}, optional) – Controls if intermediate optimization should occur. No optimization will occur if False. Defaults to False.


output – The calculation based on the Einstein summation convention.

Return type



The Einstein summation convention can be used to compute many multi-dimensional, linear algebraic array operations. einsum provides a succinct way of representing these.

A non-exhaustive list of these operations, which can be computed by einsum, is shown below along with examples:

  • Trace of an array, np.trace().

  • Return a diagonal, np.diag().

  • Array axis summations, np.sum().

  • Transpositions and permutations, np.transpose().

  • Matrix multiplication and dot product, np.matmul()

  • Vector inner and outer products, np.inner() np.outer().

  • Broadcasting, element-wise and scalar multiplication, np.multiply().

  • Tensor contractions, np.tensordot().

The subscripts string is a comma-separated list of subscript labels, where each label refers to a dimension of the corresponding operand. Whenever a label is repeated it is summed, so np.einsum('i,i', a, b) is equivalent to np.inner(a,b). If a label appears only once, it is not summed, so np.einsum('i', a) produces a view of a with no changes. A further example np.einsum('ij,jk', a, b) describes traditional matrix multiplication and is equivalent to np.matmul(a,b). Repeated subscript labels in one operand take the diagonal. For example, np.einsum('ii', a) is equivalent to np.trace(a).

In implicit mode, the chosen subscripts are important since the axes of the output are reordered alphabetically. This means that np.einsum('ij', a) doesn’t affect a 2D array, while np.einsum('ji', a) takes its transpose. Additionally, np.einsum('ij,jk', a, b) returns a matrix multiplication, while, np.einsum('ij,jh', a, b) returns the transpose of the multiplication since subscript ‘h’ precedes subscript ‘i’.

In explicit mode the output can be directly controlled by specifying output subscript labels. This requires the identifier ‘->’ as well as the list of output subscript labels. This feature increases the flexibility of the function since summing can be disabled or forced when required. The call np.einsum('i->', a) is like np.sum(a, axis=-1), and np.einsum('ii->i', a) is like np.diag(a). The difference is that einsum does not allow broadcasting by default. Additionally np.einsum('ij,jh->ih', a, b) directly specifies the order of the output subscript labels and therefore returns matrix multiplication, unlike the example above in implicit mode.

To enable and control broadcasting, use an ellipsis. Default NumPy-style broadcasting is done by adding an ellipsis to the left of each term, like np.einsum('...ii->...i', a). To take the trace along the first and last axes, you can do np.einsum('i...i', a), or to do a matrix-matrix product with the left-most indices instead of rightmost, one can do np.einsum('ij...,jk...->ik...', a, b).

When there is only one operand, no axes are summed, and no output parameter is provided, a view into the operand is returned instead of a new array. Thus, taking the diagonal as np.einsum('ii->i', a) produces a view.

The optimize argument which will optimize the contraction order of an einsum expression. For a contraction with three or more operands this can greatly increase the computational efficiency at the cost of a larger memory footprint during computation.

Typically a ‘greedy’ algorithm is applied which empirical tests have shown returns the optimal path in the majority of cases. ‘optimal’ is not supported for now.


This function differs from the original numpy.einsum in the following way(s):

  • Does not support ‘optimal’ strategy

  • Does not support the alternative subscript like

    einsum(op0, sublist0, op1, sublist1, …, [sublistout])

  • Does not produce view in any cases


>>> a = np.arange(25).reshape(5,5)
>>> b = np.arange(5)
>>> c = np.arange(6).reshape(2,3)

Trace of a matrix:

>>> np.einsum('ii', a)

Extract the diagonal (requires explicit form):

>>> np.einsum('ii->i', a)
array([ 0.,  6., 12., 18., 24.])

Sum over an axis (requires explicit form):

>>> np.einsum('ij->i', a)
array([ 10.,  35.,  60.,  85., 110.])
>>> np.sum(a, axis=1)
array([ 10.,  35.,  60.,  85., 110.])

For higher dimensional arrays summing a single axis can be done with ellipsis:

>>> np.einsum('...j->...', a)
array([ 10.,  35.,  60.,  85., 110.])

Compute a matrix transpose, or reorder any number of axes:

>>> np.einsum('ji', c)
array([[0., 3.],
       [1., 4.],
       [2., 5.]])
>>> np.einsum('ij->ji', c)
array([[0., 3.],
       [1., 4.],
       [2., 5.]])
>>> np.transpose(c)
array([[0., 3.],
       [1., 4.],
       [2., 5.]])

Vector inner products:

>>> np.einsum('i,i', b, b)

Matrix vector multiplication:

>>> np.einsum('ij,j', a, b)
array([ 30.,  80., 130., 180., 230.])
>>>, b)
array([ 30.,  80., 130., 180., 230.])
>>> np.einsum('...j,j', a, b)
array([ 30.,  80., 130., 180., 230.])

Broadcasting and scalar multiplication:

>>> np.einsum('..., ...', np.array(3), c)
array([[ 0.,  3.,  6.],
       [ 9., 12., 15.]])
>>> np.einsum(',ij', np.array(3), c)
array([[ 0.,  3.,  6.],
       [ 9., 12., 15.]])
>>> np.multiply(3, c)
array([[ 0.,  3.,  6.],
       [ 9., 12., 15.]])

Vector outer product:

>>> np.einsum('i,j', np.arange(2)+1, b)
array([[0., 1., 2., 3., 4.],
       [0., 2., 4., 6., 8.]])

Tensor contraction:

>>> a = np.arange(60.).reshape(3,4,5)
>>> b = np.arange(24.).reshape(4,3,2)
>>> np.einsum('ijk,jil->kl', a, b)
array([[4400., 4730.],
       [4532., 4874.],
       [4664., 5018.],
       [4796., 5162.],
       [4928., 5306.]])

Example of ellipsis use:

>>> a = np.arange(6).reshape((3,2))
>>> b = np.arange(12).reshape((4,3))
>>> np.einsum('ki,jk->ij', a, b)
array([[10., 28., 46., 64.],
       [13., 40., 67., 94.]])
>>> np.einsum('ki,...k->i...', a, b)
array([[10., 28., 46., 64.],
       [13., 40., 67., 94.]])
>>> np.einsum('k...,jk', a, b)
array([[10., 28., 46., 64.],
       [13., 40., 67., 94.]])

Chained array operations. For more complicated contractions, speed ups might be achieved by repeatedly computing a ‘greedy’ path. Performance improvements can be particularly significant with larger arrays:

>>> a = np.ones(64).reshape(2,4,8)
# Basic `einsum`: ~42.22ms  (benchmarked on 3.4GHz Intel Xeon.)
>>> for iteration in range(500):
...     np.einsum('ijk,ilm,njm,nlk,abc->',a,a,a,a,a)
# Greedy `einsum` (faster optimal path approximation): ~0.117ms
>>> for iteration in range(500):
...     np.einsum('ijk,ilm,njm,nlk,abc->',a,a,a,a,a, optimize=True)