Dummy Index (Einstein Notation)

linear-algebra

Definition

Dummy Index (Einstein Notation)

A dummy index (contracted index) is an index that appears exactly twice in an Einstein notation term. It is implicitly summed over its full range and does not appear in the output.

Examples

Example

In $A_{ij}{B}_{jk}$, the dummy index is $j$. It is summed from $1$ to $n$ and removed — the output retains only $i$ and $k$.

Each dummy index reduces the output dimensionality by two relative to the product of the inputs (one axis from each factor).