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TensorFlow 1 version View source on GitHub

A generalized contraction between tensors of arbitrary dimension.



Used in the guide:

Used in the tutorials:

This function returns a tensor whose elements are defined by equation, which is written in a shorthand form inspired by the Einstein summation convention. As an example, consider multiplying two matrices A and B to form a matrix C. The elements of C are given by:

  C[i,k] = sum_j A[i,j] * B[j,k]

The corresponding equation is:


In general, the equation is obtained from the more familiar element-wise equation by 1. removing variable names, brackets, and commas, 2. replacing "*" with ",", 3. dropping summation signs, and 4. moving the output to the right, and replacing "=" with "->".

Many common operations can be expressed in this way. For example:

# Matrix multiplication
>>> einsum('ij,jk->ik', m0, m1)  # output[i,k] = sum_j m0[i,j] * m1[j, k]

# Dot product
>>> einsum('i,i->', u, v)  # output = sum_i u[i]*v[i]

# Outer product
>>> einsum('i,j->ij', u, v)  # output[i,j] = u[i]*v[j]

# Transpose
>>> einsum('ij->ji', m)  # output[j,i] = m[i,j]

# Trace
>>> einsum('ii', m)  # output[j,i] = trace(m) = sum_i m[i, i]

# Batch matrix multiplication
>>> einsum('aij,ajk->aik', s, t)  # out[a,i,k] = sum_j s[a,i,j] * t[a, j, k]

To enable and control broadcasting, use an ellipsis. For example, to do batch matrix multiplication, you could use:

einsum('...ij,...jk->...ik', u, v)

This function behaves like numpy.einsum, but does not support:

  • Subscripts where an axis appears more than once for a single input (e.g. ijj,k->ik) unless it is a trace (e.g. ijji).


  • equation: a str describing the contraction, in the same format as numpy.einsum.
  • *inputs: the inputs to contract (each one a Tensor), whose shapes should be consistent with equation.
  • name: A name for the operation (optional).


The contracted Tensor, with shape determined by equation.


  • ValueError: If
    • the format of equation is incorrect,
    • the number of inputs implied by equation does not match len(inputs),
    • an axis appears in the output subscripts but not in any of the inputs,
    • the number of dimensions of an input differs from the number of indices in its subscript, or
    • the input shapes are inconsistent along a particular axis.