Free Index (Einstein Notation)

linear-algebra

Definition

Free Index (Einstein Notation)

A free index is an index that appears exactly once in an Einstein notation term. It is not summed over and becomes an axis of the output.

Examples

Example

In $A_{ij}{B}_{jk}$, the free indices are $i$ and $k$. The output has shape $m\times p$ because $i\in [1,m]$ and $k\in [1,p]$.

The set of free indices determines the output type: zero free indices → scalar, one → vector, two → matrix, and so on.