13.2.4 Tensor Full Contraction Area

Tensor Full Contraction Area involves summing all tensor indices to yield a scalar value through complete contraction.

Tensor Full Contraction Area is the domain within tensor contraction areas concerned with contractions carried out until every available index of a tensor has been paired and summed, producing an order-zero scalar and representing the complete exhaustion of a tensor's index structure through repeated application of the contraction operation.

Defining the Full Contraction Case

Complete Exhaustion of Index Slots

A full contraction area applies when the number of independent contractions performed equals exactly half the original order of the tensor, so that every contravariant index has been paired with a covariant index and no free indices remain in the result.

count (contractions) = \frac{order (T)}{2}

Requirement of an Even, Balanced Order

Full contraction area applies only to tensors possessing an equal number of contravariant and covariant indices, since an imbalance between the two counts would leave at least one index of the majority type unpaired regardless of how the available contractions are arranged.

count (upper) = count (lower)

The Resulting Scalar

Order Reduced Entirely to Zero

The characteristic outcome of full contraction area is a result of order zero, since every index originally present has been consumed by one of the contractions applied.

order (result) = 0

Complete Invariance Under Change of Basis

The scalar produced by full contraction is entirely unchanged under any change of basis, a property that follows from the tensor transformation law once every index has been eliminated, since no transformation matrix remains to act upon a result with no indices.

\tilde{c} = c

Representative Instances of Full Contraction

Contracting a Rank-Two Tensor Entirely

A tensor possessing exactly one upper and one lower index is fully contracted through a single summation, directly yielding a scalar without any intermediate partial results.

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

Contracting a Higher-Order Tensor Through Repeated Pairing

A tensor with two upper and two lower indices reaches full contraction area only once both available pairs have been summed, whether performed as two sequential single contractions or as one combined multiple contraction, since both approaches consume all four indices.

T_{i j}^{i j}

Verification Specific to Full Contraction

Simplified Invariance Check

Because the result of a full contraction is a scalar, invariance verification reduces to confirming exact numerical equality of the computed value across two different bases, rather than confirming agreement with a nontrivial transformation formula involving surviving free indices.

Confirming No Index Remains Unpaired

Verification within the full contraction area includes explicitly confirming that no index of the original tensor has been left out of every applied contraction, since an overlooked index would leave the result with a nonzero order and place it outside this area despite an intention to contract fully.

Distinction from Partial Contraction

Full contraction area is distinguished from partial contraction area by the complete absence of any surviving free index, marking the endpoint that partial contraction approaches but does not reach, and representing the maximal extent to which the contraction operation can reduce a tensor's order.

Relationship to Tensor Operation Notation

Full contraction area is signaled in tensor operation notation by an expression in which every index symbol present is repeated as a matched upper-lower pair, with no index symbol appearing only once, so that the total absence of any singly occurring index identifies an expression as denoting a fully contracted, order-zero result.