13.14.2 Tensor Contraction Contravariant Count Change

Tensor Contraction Contravariant Count Change is the specific decrease undergone by the contravariant index count component of a tensor's type as a direct consequence of contraction, tracked in isolation from the parallel change occurring in the covariant count. It isolates one half of the overall type reduction produced by contraction, focusing exclusively on how the number of upper indices evolves as successive contractions are applied, independent of any accompanying discussion of the covariant side.

Conceptual Basis

Isolating One Component of the Type Pair

A tensor's type is described by two counts considered jointly, but the contravariant count change examines only the first of these two numbers, tracking how it decreases as contractions proceed, without folding in the simultaneous behavior of the covariant count into the same discussion.

Why the Contravariant Count Decreases

Each valid contraction, by requiring one contravariant index to be consumed alongside one covariant index, necessarily reduces the contravariant count by exactly one for every contraction performed, making the change entirely predictable and directly tied to the number of contractions applied.

Relationship to Which Contravariant Slots Are Consumed

The contravariant count change records only the numerical reduction in count, not which specific contravariant slots were selected for contraction; different choices of contravariant slot pair selection can produce the same numerical count change while consuming different particular indices.

Formal Description

General Formula for the Count Change

For a tensor with original contravariant count $$p$$ subjected to $$k$$ contractions, the resulting contravariant count is:

with the change itself, the amount by which the count has decreased, equal to $$k$$.

Bound on the Possible Change

Since each contraction requires an available covariant index as well, the contravariant count change is bounded by the original covariant count $$q$$, meaning:

so that the contravariant count cannot be reduced below $$p-\min (p,q)$$ regardless of how many contractions are attempted.

Independence From Which Specific Indices Are Chosen

If a tensor has several contravariant indices available, contracting any one of them produces the same magnitude of contravariant count change, namely a decrease of one, regardless of which specific slot among the several candidates was selected for the contraction.

Properties

Direct Proportionality to Contraction Count

The contravariant count change is always exactly equal to the number of contractions performed, establishing a direct and simple proportional relationship between these two quantities that holds regardless of the tensor's rank or the specific contractions chosen.

Zero Change Under Contractions Not Involving Contravariant Slots

If an operation performed on a tensor does not involve summing over any contravariant index, such as certain symmetrization procedures that do not constitute contraction, the contravariant count change associated with that operation is zero, distinguishing genuine contraction from other rank-preserving or structure-altering operations.

Coupling With the Covariant Count Change

Although considered in isolation here, the contravariant count change is always numerically equal to the covariant count change for any valid contraction, since both counts decrease by the same amount with each contraction performed, a coupling that follows directly from the requirement that contraction always removes one index of each variance.

Applications

Tracking Progress Toward a Scalar Result

Monitoring the contravariant count change across a sequence of contractions provides a direct way of tracking progress toward the scalar result case, since this case is reached precisely when the contravariant count has been reduced to zero, alongside the covariant count.

Predicting the Type of Intermediate Tensors

In a multi-step computation, knowing the contravariant count change expected from each planned contraction allows the type of every intermediate tensor in the sequence to be predicted in advance, aiding in the design and verification of complex tensor expressions.

Diagnostic Use in Verifying Contraction Counts

Comparing the observed contravariant count change against the number of contractions actually performed provides a straightforward consistency check, since any discrepancy would indicate that an operation other than a standard contraction, or an incorrectly specified contraction, has been applied.