13.4.2 Tensor Single Contraction Summation Index

Tensor Single Contraction Summation Index sums over one index to simplify tensor operations, enabling concise notation in multilinear algebra.

Tensor Single Contraction Summation Index is the shared index symbol assigned to the two members of a single contraction slot pair, serving as the dummy variable that ranges over every value of the common dimension while the components of the tensor are summed during the single index contraction operation.

The Nature of a Summation Index

A Dummy Variable Rather Than a Fixed Label

The summation index does not refer to any single fixed value but instead stands for each value in turn as the summation proceeds, distinguishing it from a free index, which denotes a specific, though unspecified, fixed position in the resulting tensor.

T_{i}^{i} = \sum_{i = 1}^{n} T_{i}^{i}

Absence from the Final Result

Because the summation index is summed away entirely, it does not appear in the resulting tensor's index list, distinguishing the summation index from any free index of the input, which persists in the output with its original label available for reference.

Range of the Summation Index

Determined by the Common Dimension

The summation index ranges over every integer value from one up to the dimension shared by the two slots of the contraction pair, since both slots of a valid pair are required to agree in dimension for the contraction to be defined.

i = 1, 2, \dots, n

Consequence of a Mismatched Dimension

If the two slots selected for a contraction were to disagree in dimension, no single consistent range could be assigned to a shared summation index, which is precisely why equal dimension between the two slots is required before the summation index can be meaningfully introduced.

Naming and Choice of the Summation Index Symbol

Arbitrariness of the Chosen Letter

The particular letter chosen to denote the summation index carries no meaning beyond identifying which two slots are linked, so replacing the summation index with any other unused symbol throughout the expression leaves the value of the contraction unchanged.

T_{i}^{i} = T_{m}^{m}

Requirement of a Unique Symbol Within Its Scope

Within a single application of the operation, the summation index symbol must not coincide with any other index symbol appearing elsewhere in the same term with a different intended role, since reusing a symbol ambiguously would obscure which positions are meant to be linked by that particular summation.

The Summation Index in Sequences of Contractions

Independent Symbols for Independent Summations

When several single index contraction operations are applied together, each is assigned its own summation index symbol, distinct from the others, so that the several summations remain clearly separated and independently interpretable within the combined expression.

A_{i} B_{j}^{i} C^{j}

Reuse of a Symbol Across Separate Terms

A summation index symbol used in one term of a larger sum may be reused with an entirely independent meaning in a separate term of the same sum, since the summation convention confines each summation index to the single term in which it appears.

Relationship to Tensor Operation Notation

The summation index is represented in tensor operation notation by the repeated appearance of a single symbol as both an upper and lower index within one term, with the summation convention supplying the understanding that this repeated symbol denotes a dummy variable ranging over the full dimension, rather than a fixed or free index of the resulting expression.