13.17.3 Tensor Contraction Sequence Associativity Context

Tensor contraction sequence associativity explores how repeated index contractions in tensor algebra maintain consistent results regardless of grouping order.

Tensor Contraction Sequence Associativity Context is the set of conditions under which a multi-step contraction sequence can be regrouped, so that the order in which intermediate products are formed and contracted is changed while the final residual tensor remains identical, mirroring the associative property familiar from ordinary multiplication but applied to chains of tensor contractions.

Definition

For a chain of tensors $A, B, C$ linked by contractions, the associativity context asserts that grouping the contractions as $(A B) C$ or as $A (B C)$ produces the same final tensor, provided the same underlying index pairs are contracted in both groupings:

(A_{a}^{i} B_{b}^{a}) C_{j}^{b} = A_{a}^{i} (B_{b}^{a} C_{j}^{b})

Conditions for Associativity

Same Underlying Contractions

Associativity of a contraction sequence holds when regrouping does not change which specific index pairs are contracted, only the order in which the intermediate partial products are assembled and summed. The regrouping is purely a matter of computational strategy, not a change to the mathematical operation being performed.

Consistency With Summation Interchange

The associativity context relies on the fact that finite sums can be reordered and regrouped freely, so that summing over index $a$ first and then over $b$ , or vice versa, produces the same total:

\sum_{b} (\sum_{a} A_{a}^{i} B_{b}^{a}) C_{j}^{b} = \sum_{a} A_{a}^{i} (\sum_{b} B_{b}^{a} C_{j}^{b})

which holds by finiteness of the index ranges and the standard distributive and commutative properties of addition and multiplication of scalars.

Practical Consequences

Computational Strategy Choice

Because the final result does not depend on grouping, an implementer of a contraction sequence is free to choose whichever intermediate grouping minimizes computational cost, such as forming the smaller intermediate product first, without affecting correctness.

Diagram Invariance

In the diagrammatic representation, associativity context corresponds to the fact that a chain of nodes joined by edges represents a single well-defined tensor regardless of which sub-chain of edges is conceptually evaluated first; the diagram itself carries no notion of grouping order.

Visual Comparison of Two Groupings

Both groupings produce the identical diagram, and therefore the identical residual tensor.

Boundary of the Context

Associativity context applies strictly to regrouping of contraction order, not to reassignment of which indices are contracted; changing which index pair is paired in a contraction produces, in general, a genuinely different tensor and falls outside the scope of this associativity context.