13.14.3 Tensor Contraction Total Order Change

Tensor Contraction Total Order Change is the decrease in a tensor's overall order, understood as the combined total of its contravariant and covariant index counts, that results from contraction, expressed as a single aggregate number rather than as the separate contravariant and covariant changes considered individually. It captures the net reduction in the total number of indices a tensor carries, providing a coarser but more immediately interpretable measure of how much a contraction has simplified an object's index structure.

Conceptual Basis

Order as the Sum of Both Index Counts

The order of a tensor, sometimes called its total rank, is the sum of its contravariant and covariant counts taken together, without distinguishing between the two variances. The total order change tracks how this single combined number decreases as a result of contraction, rather than tracking the contravariant and covariant contributions separately.

Why a Combined Measure Is Useful

While the separate contravariant and covariant count changes provide detailed information about the type of the resulting tensor, the total order change offers a simpler, single-number summary of how much an object has been simplified, useful in contexts where the fine distinction between variances is less immediately relevant than the overall complexity of the tensor.

Direct Consequence of the Pairing Requirement

Because every valid contraction removes exactly one contravariant and one covariant index together, the total order necessarily decreases by exactly two with each contraction performed, making the total order change a direct and entirely predictable multiple of the number of contractions applied.

Formal Description

General Formula

For a tensor of original order $$p+q$$ subjected to $$k$$ contractions, the resulting order is:

so that the total order change itself is:

Relationship to the Individual Count Changes

Because the contravariant count change and the covariant count change are each equal to $$k$$ individually, the total order change is simply their sum, confirming that the aggregate measure is fully consistent with, and derivable from, the finer-grained changes in each separate count.

Bound on the Total Order Change

Since $$k$$ cannot exceed $$\min (p,q)$$, the total order change is similarly bounded:

with equality precisely when the scalar result case is reached.

Properties

Always an Even Number

Because each contraction removes indices in matched pairs, the total order change is always an even number, equal to twice the number of contractions performed, regardless of how those contractions are distributed between the contravariant and covariant sides.

Insufficiency to Determine the Resulting Type Alone

While the total order change indicates how much simpler the resulting tensor is overall, it does not by itself specify the resulting type, since the same total order change could in principle correspond to different combinations of remaining contravariant and covariant counts if the original tensor's counts were unequal.

Additivity Across Successive Contractions

If a tensor undergoes several contractions in succession, the total order change accumulated over the entire sequence equals the sum of the total order changes contributed by each individual contraction, consistent with the additive nature of counting contractions.

Applications

Quick Assessment of Simplification

The total order change provides a fast, single-number way of assessing how much a tensor expression has been simplified by a given sequence of contractions, useful for comparing the relative complexity of different computational strategies without needing to track contravariant and covariant counts separately.

Estimating Computational Cost

In numerical tensor computation, the total order change relates directly to how much the storage and computational requirements of an object have been reduced, since fewer total indices generally correspond to smaller arrays and less costly subsequent operations.

Complementary Use Alongside Type Tracking

While the total order change is useful as a summary measure, it is typically used alongside, rather than instead of, tracking of the full type, since the type retains the additional information about the balance between contravariant and covariant counts that the aggregate order change alone does not capture.