13.5 Tensor Multiple Index Contraction Operation

Tensor Multiple Index Contraction Operation is the combined application of two or more single index contraction operations to the same tensor or product of tensors, each summing over its own independently selected pair of contravariant and covariant indices, treated together as a single composite operation.

Composition from Elementary Operations

Combining Several Slot Pair Selections

The multiple index contraction operation begins by performing slot pair selection more than once, identifying several distinct pairs of contravariant and covariant slots on the input, with each pair assigned its own distinct summation index symbol.

Independent Summation Over Each Pair

Once the several pairs are selected, the operation carries out index summation separately for each pair, with the summations over distinct pairs proceeding without interference, since each involves a different summation index ranging independently over its own dimension.

Effect on Type and Order

Cumulative Type Reduction

Each of the several pairs contracted reduces both the contravariant count and the covariant count of the input's type by one, so that a multiple index contraction operation involving $k$ pairs produces a result of type $(p-k,q-k)$.

Cumulative Order Reduction

Correspondingly, the total order of the result is reduced by twice the number of pairs contracted, matching the sum of the fixed reduction of two contributed by each of the individual single index contraction operations composing the whole.

Requirements Governing the Combined Operation

Disjointness of the Selected Slot Pairs

The multiple index contraction operation requires that no single slot be claimed by more than one of the selected pairs, since a slot committed to two different pairings simultaneously would leave the intended summation ambiguous.

Independent Satisfaction of Pairwise Requirements

Every pair among those selected must independently satisfy the requirements governing a single contraction, namely opposite variance and equal dimension between its two members, since the validity of one pair provides no assurance regarding the validity of any other pair within the same combined operation.

Order of Application Among the Component Pairs

Commutativity of Independent Summations

Because each contracted pair involves its own distinct summation index acting on positions not shared with any other pair, the several single index contraction operations composing the whole may be carried out in any order, or effectively simultaneously, without altering the final result.

Equivalence to a Sequence of Single Contractions

A multiple index contraction operation applied to a set of disjoint pairs produces exactly the same result as applying the corresponding single index contraction operations one after another in sequence, confirming that the multiple index operation is a composite built entirely from repeated application of the elementary operation.

Relationship to Tensor Operation Notation

The multiple index contraction operation is denoted in tensor operation notation by an expression containing more than one distinct index symbol, each repeated exactly once as a matched upper and lower pair, with the number of such distinct repeated symbols indicating precisely how many single index contraction operations have been combined within the expression.