Einsum Notation

linear-algebra

Definition

Einsum Notation

Einstein summation notation is a concise way to represent summations over repeated indices when performing tensor operations. Instead of writing out explicit summation symbols , we reuse an index letter in multiple arrays to imply a summation over that index. This allows for compact expressions of operations like matrix multiplication, tensor contraction, outer product, etc.

Understanding

Let and be two matrices:

To compute , we need to replace the repeated indices . Since occurs on the left-hand side of the arrow and not on the right-hand side, we need to sum over for each pair.

The shape of the resulting tensor is equal to the dimensionalities of the indices and , thus and respectively:

Now, we need to compute each element by summing over the index:

Thus, the resulting matrix is:

Note that this is equivalent to matrix multiplication, which is a special case of tensor contraction.

Examples

The examples are taken from ¹.

Transpose

The transpose of a matrix is denoted as:

or equivalently:

einsum('ij->ji', A)

Sum

The sum of all elements of a matrix is denoted as:

or equivalently:

einsum('ij->', A)

Column Sum

The sum of all elements of the -th column of a matrix is denoted as:

or equivalently:

einsum('ij->j', A)

Row Sum

The sum of all elements of the -th row of a matrix is denoted as:

or equivalently:

einsum('ij->i', A)

Dot Product

The dot product of two vectors is denoted as:

einsum('i,i->', x, y)

Footnotes

1. https://rockt.ai/2018/04/30/einsum ↩