13.12.4 Tensor Covariant Contravariant Type Reduction

Tensor Covariant Contravariant Type Reduction is the change in a tensor's type, expressed as the pair of contravariant and covariant index counts, that results specifically from applying the covariant contravariant contraction case, with each contraction of one matched pair lowering both counts by exactly one. It describes how the classification of a tensor by its number of upper and lower indices evolves as successive contractions are applied, tracking this evolution as a direct numerical consequence of the underlying contraction rule.

Conceptual Basis

Type as the Object of Reduction

A tensor's type, written as a pair of counts $$(p,q)$$ for contravariant and covariant indices respectively, is the quantity being reduced. Type reduction refers specifically to how this pair changes as contractions satisfying the covariant contravariant requirement are carried out.

Symmetric Reduction of Both Counts

Because a valid contraction always removes exactly one contravariant index and one covariant index together, type reduction is inherently symmetric: the two counts in the type pair decrease by the same amount with every contraction performed, never independently of one another.

Reduction Bounded by the Smaller Count

Since each contraction requires both a contravariant and a covariant index to consume, the total number of contractions possible is limited by whichever of the two original counts is smaller, meaning type reduction cannot proceed indefinitely once one variance type is exhausted.

Formal Description

General Reduction Formula

For a tensor of original type $$(p,q)$$ subjected to $$k$$ contractions each satisfying the covariant contravariant requirement, the resulting type is:

valid for any $$k$$ satisfying $$0\le k\le \min (p,q)$$.

Boundary Case of Full Reduction

When $$k=\min (p,q)$$, the type reduction reaches its maximum extent, leaving a tensor of type:

which, whenever $$p=q$$, coincides with the scalar type $$(0,0)$$.

Invariance of the Type Difference

Throughout any sequence of contractions satisfying the covariant contravariant requirement, the difference $$p-q$$ remains unchanged, since both counts are reduced by the same amount at every step, making this difference a conserved quantity under type reduction.

Properties

Path Independence of the Final Type

Regardless of the specific order in which individual matched pairs are contracted, applying the same total number of contractions to a tensor produces a resulting object of the same type, since type reduction depends only on the count of contractions performed, not their sequence.

Distinguishing Type Reduction From Rank Reduction

While rank, understood as the total index count $$p+q$$, decreases by two with each contraction, type reduction additionally records how this decrease is distributed between the contravariant and covariant counts specifically, providing finer information than rank alone.

Governing Further Contractibility

The reduced type directly determines whether additional contractions remain possible, since any further contraction requires both a nonzero contravariant count and a nonzero covariant count in the current type.

Applications

Predicting the Outcome of Planned Contractions

Before performing a sequence of contractions on a tensor, applying the type reduction formula in advance allows the type, and therefore the essential character, of the final result to be predicted without carrying out the full computation.

Classifying Intermediate Results in Multi-Step Computations

In a computation involving several successive contractions, tracking the reduced type at each stage provides a systematic way of classifying and verifying the intermediate tensors produced, ensuring that each step behaves as expected given the number of contractions applied so far.

Connecting to Named Contraction Cases

The type reduction associated with trace contraction, matrix multiplication, and vector covector pairing can each be understood as specific instances of the general reduction formula, differing only in the particular values of $$p$$, $$q$$, and $$k$$ involved.