Contraction (Einstein Notation)

linear-algebra

Definition

Contraction (Einstein Notation)

A contraction is the summation over a dummy index shared by two factors in an Einstein notation expression.

Examples

Example

In $A_{ij}{x}_j$, the index $j$ is contracted: it appears in both $A$ and $x$ and is summed from $1$ to $n$. The output inherits only the free index $i$.

A contraction removes one axis from each participating factor. After contraction, the output has two fewer axes than the combined inputs.