13.6 Tensor Full Contraction Operation

Tensor Full Contraction Operation is the composite procedure of applying contraction repeatedly to a tensor possessing an equal number of contravariant and covariant indices until every one of its slots has been paired and summed, yielding a scalar as the final output of the combined procedure.

Structure of the Operation

A Complete Slot Pair Set

The full contraction operation is defined by a slot pair set whose members together account for every index slot of the input tensor, leaving no contravariant or covariant position outside of some pair.

Realization as a Multiple Index Contraction

Because full contraction generally involves more than one pair whenever the input's order exceeds two, the operation is realized as a particular instance of the multiple index contraction operation, one in which the slot pair set happens to exhaust every available slot rather than leaving any free indices behind.

Precondition for the Operation

Equal Contravariant and Covariant Counts

The full contraction operation requires that the input tensor's type consist of equal contravariant and covariant counts, since only under this balance can every slot be assigned to some pair without leaving an index of the majority category unmatched.

Exhaustive Coverage of All Slots

Beyond the balance of counts, the operation requires that the specific slot pair set chosen genuinely covers every slot of the tensor, since a balanced type alone does not guarantee that a particular chosen set of pairings leaves nothing free, if some slots were mistakenly left out of every pair.

Outcome of the Operation

A Scalar of Order Zero

Applying the full contraction operation reduces the input tensor's order entirely to zero, producing a single value with no remaining index structure, since every original index has been consumed by its assigned pair.

Invariance of the Resulting Scalar

The scalar produced by the operation remains entirely unchanged under any change of basis, a property following directly from the general invariance guaranteed for any tensor's components once every index has been eliminated through valid contraction.

Carrying Out the Operation in Practice

Sequential Application of Elementary Contractions

The full contraction operation can be carried out as a sequence of single index contraction operations, each consuming one pair from the slot pair set, applied one after another until no slots remain, with the final result independent of the particular sequence chosen among the disjoint pairs.

Direct Evaluation as a Single Nested Summation

Alternatively, the operation may be evaluated directly as one combined nested summation over the entire summation set associated with the chosen slot pair set, computing the scalar result in a single combined step rather than through a visibly sequential process.

Distinguishing the Operation from Partial Contraction

The full contraction operation is distinguished from an operation that applies only some of the available pairings by the exhaustive coverage of its slot pair set, with any omission of even one available pair placing the resulting procedure instead within the domain of partial contraction rather than full contraction.

Relationship to Tensor Operation Notation

The full contraction operation is signaled in tensor operation notation by an expression in which every index symbol present is repeated as a matched upper-lower pair, with the complete absence of any singly occurring index symbol confirming that the notated expression denotes an operation exhausting the entire index structure of the tensor involved.