13.3.4 Tensor Contraction Type Reduction

Tensor Contraction Type Reduction is the change in a tensor's type, understood as the paired counts of its contravariant and covariant indices, that results from applying the contraction operation, describing how the specific numbers of upper and lower indices decrease as a consequence of the operation rather than merely how the total order decreases.

Type as Distinct from Order

The Type of a Tensor

A tensor's type is expressed as a pair of counts, one for its contravariant indices and one for its covariant indices, providing more detailed information than the total order alone, since two tensors can share the same order while differing in how that order is distributed between upper and lower indices.

Here $p$ denotes the number of contravariant indices and $q$ denotes the number of covariant indices.

Why Contraction Affects Type Specifically

Because contraction removes exactly one contravariant index and one covariant index, its effect is most precisely described in terms of the type pair rather than the single order value, since the reduction is not distributed arbitrarily but falls specifically and equally on both index categories.

The Reduction Formula

Decrease in Both Components of the Type

A single contraction reduces both the contravariant count and the covariant count of a tensor's type by exactly one, transforming a tensor of type $(p,q)$ into a tensor of type $(p-1,q-1)$.

Consequence for the Order

Because order is defined as the sum of the two type components, the reduction of each component by one produces the familiar decrease of two in the total order, so the order reduction associated with contraction follows directly from the more detailed type reduction.

Type Reduction Under Multiple Contractions

Cumulative Reduction Across Independent Pairings

When several independent contractions are applied, each reduces both components of the type by one, so that the type after $k$ independent contractions becomes $(p-k,q-k)$.

Reaching Type Zero Through Full Contraction

When the original contravariant and covariant counts are equal, repeated contraction can reduce the type entirely to $(0,0)$, the type associated with a scalar, matching the endpoint described by tensor scalar result scope.

Bound on Achievable Type Reduction

Limitation by the Smaller Index Count

Because each contraction requires one contravariant and one covariant index, the maximum number of contractions applicable to a tensor is limited by whichever of the two original counts is smaller, so type reduction cannot proceed further once one of the two components reaches zero.

Residual Type When Counts Differ

If the original contravariant and covariant counts are unequal, full contraction on all possible pairs still leaves a nonzero type, with the surplus indices of the majority category persisting as free indices in the result.

Relationship to Tensor Operation Notation

Type reduction is observed in tensor operation notation by comparing the count of distinct upper index symbols and distinct lower index symbols in an expression before and after accounting for repeated pairs, since each matched upper-lower repetition removes exactly one symbol from each count, directly reflecting the reduction applied to the tensor's type.