Ill-formed Index (Einstein Notation)

linear-algebra

Definition

Ill-formed Index (Einstein Notation)

In Einstein notation, an index that appears three or more times in a single term is ill-formed. The expression is undefined because the summation is ambiguous — there is no rule for which pairs to contract.

Every index in a well-formed term must be either free (once) or dummy (twice).

Examples

$A_{ij}{B}_{jk}{x}_j$

The index $j$ appears three times (in $A$, $B$, and $x$). It is unclear whether to sum $A$ with $B$, $A$ with $x$, or $B$ with $x$.