8.8.1 Tensor Repeated Index Pair Pattern

The Tensor Repeated Index Pair Pattern denotes contraction and summation in tensor algebra through paired index notation.

Tensor Repeated Index Pair Pattern is the broader organizational pattern describing how multiple upper-lower index pairs can be distributed, combined, and arranged throughout a tensor expression, governing how many separate pairs an expression contains, how those pairs relate to the free indices also present, and how the overall pattern of pairing determines the structure of the final result.

Enumerating Pairs Within an Expression

Counting Distinct Pairs

Any tensor expression can be analyzed by counting how many distinct upper-lower pairs it contains, with each pair identified by a unique index letter occurring once upper and once lower, and this count directly determines how many separate summations are implicitly performed.

A_{i j} B^{j k} C_{k}

This expression contains two distinct pairs, one formed by (j) linking (A) and (B), and another formed by (k) linking (B) and (C), and identifying both pairs correctly is necessary to determine that the final result carries only the free index (i).

Chained Pairs

A characteristic arrangement within the pair pattern is the chain, in which one tensor factor shares a pair with its left neighbor and a separate pair with its right neighbor, linking a sequence of factors together much as matrix multiplication chains several matrices through shared inner dimensions.

Interaction Between Pairs and Free Indices

Pairs Coexisting with Free Indices

The pair pattern of an expression is only ever part of the full index picture; any indices not absorbed into a pair remain free, and the pair pattern must be identified precisely in order to separate correctly which indices vanish through summation and which survive into the output.

R_{i} = A_{i j} B^{j}

Disjoint Pairs Within a Single Term

A single term can contain multiple pairs that do not share any tensor factor in common, each pair operating entirely independently of the others, contributing separate, unrelated summations that are simply multiplied together within the same term.

A_{i} B^{i} C_{j} D^{j}

Pattern Recognition Across Longer Expressions

Reading a Complex Expression Systematically

Analyzing the pair pattern of a long expression proceeds by scanning for every index letter, tallying its occurrences and their variance, grouping letters that form valid upper-lower pairs, and setting aside letters that occur only once as free indices, which together give a complete structural map of the expression.

Errors Revealed by the Pattern

Examining the pair pattern also exposes malformed expressions directly: a letter occurring twice with the same variance, or three or more times overall, breaks the expected pattern and signals that the expression requires correction before its meaning can be considered well defined.

Practical Illustration

Mapping the full pair pattern of an expression, rather than examining indices in isolation, is what allows a reader to correctly determine the rank, the surviving free indices, and the chain of contractions represented by even a lengthy tensor expression involving many factors and several simultaneous pairs.