13.5.2 Tensor Multiple Contraction Summation Set

The Tensor Multiple Contraction Summation Set simplifies tensor expressions by summing over multiple contracted indices in algebra.

Tensor Multiple Contraction Summation Set is the complete collection of distinct summation indices introduced by a multiple index contraction operation, together with the individual ranges each index is summed over, forming the full set of dummy variables whose combined summation defines the operation's result.

Composition of the Summation Set

One Summation Index per Contracted Pair

For every slot pair selected within a multiple index contraction operation, the summation set contains exactly one corresponding summation index, so that the size of the set equals the number of independent pairs being contracted.

{i, j}

Distinctness of the Members

Every member of the summation set is a distinct symbol, since assigning the same symbol to two different pairs would conflate the two contractions, so the requirement of distinctness among the pairs directly implies the distinctness of the corresponding elements within the summation set.

T_{i j}^{i j}

The Joint Summation Defined by the Set

Nested Summation Across All Members

The full summation associated with the operation is expressed as a nested sum ranging over every member of the summation set, with each nested summation contributing its own contracted pair's effect to the overall result.

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

Range Assigned to Each Member

Each member of the summation set carries its own range, determined by the dimension shared between the two slots of its corresponding pair, with different members of the set potentially having different ranges if the underlying vector spaces involved differ in dimension.

i = 1, \dots, m; j = 1, \dots, n

The Summation Set and Total Computational Effort

Product of Ranges Determining Total Terms

The total number of terms combined by the joint summation over the entire summation set is given by the product of the ranges of its individual members, together with the number of distinct combinations of any surviving free indices, directly connecting the summation set to the concerns addressed within tensor contraction cost area.

total terms = \prod_{k \in set} range (k)

Growth of the Set with Additional Pairs

Each additional pair contracted within the operation adds one further member to the summation set, so that a larger summation set corresponds directly to a greater number of independent contractions being performed simultaneously.

The Summation Set as Distinct from Free Indices

Membership Confined to Contracted Pairs

The summation set contains only those index symbols that are summed away, explicitly excluding any free index symbol that survives into the result tensor, since free indices are not dummy variables and do not participate in the joint summation the set describes.

Complementary Relationship to the Result Structure

The summation set and the free indices comprising the result structure together account for every index symbol present in the original expression, with the summation set identifying what is removed and the result structure identifying what remains.

Relationship to Tensor Operation Notation

The summation set corresponds directly to the collection of distinct index symbols appearing as matched upper-lower pairs within tensor operation notation, with the number and identity of these repeated symbols in the notated expression fully determining the membership of the summation set associated with a given multiple index contraction operation.