13.7.3 Tensor Partial Contraction Residual Type

Tensor Partial Contraction Residual Type is the tensor type, expressed as the pair of remaining contravariant and covariant index counts, that characterizes the object produced once a partial contraction has removed some but not all of the index pairs from an original tensor. It specifies precisely how many upper and how many lower slots survive the contraction, thereby fixing the transformation behavior and algebraic classification of the resulting tensor.

Conceptual Basis

Type as a Pair of Counts

A tensor's type is conventionally written as $$(p,q)$$, where $$p$$ counts contravariant indices and $$q$$ counts covariant indices. The residual type is this same pair applied to the tensor that remains after contraction, rather than to the original tensor.

Why a Residual Type Is Needed

Because partial contraction changes the number of free indices without necessarily eliminating both variances equally, the resulting object generally has a different type from the original tensor, and this new type must be tracked explicitly to know how the residual tensor transforms and how it may be combined with other tensors.

Distinguishing Residual Type From Rank

Rank alone, understood as the total number of free indices, does not capture how many of those indices are contravariant versus covariant. The residual type retains this finer distinction, which matters because contravariant and covariant slots transform by inverse relationships under a change of basis.

Formal Description

Computing the Residual Type

If a tensor of original type $$(p,q)$$ undergoes a partial contraction removing $$k$$ matched pairs, the residual type is:

with the requirement $$0<k<\min (p,q)$$ for the residual to remain a genuine nonscalar tensor rather than a fully contracted scalar.

Example Computation

A tensor of type $$(3,2)$$ subjected to a single partial contraction, so $$k=1$$, has residual type:

reflecting two remaining upper indices and one remaining lower index.

Asymmetric Reduction Bound

Since each contraction removes exactly one contravariant and one covariant index together, the residual type is always reached by reducing $$p$$ and $$q$$ by the same amount, meaning the difference $$p-q$$ is invariant across any sequence of partial contractions.

Properties of the Residual Type

Invariance of the Type Difference

Because contraction always removes matched pairs, the quantity $$p-q$$ computed from the residual type always equals the same quantity computed from the original tensor's type, regardless of how many contractions have been performed.

Determining Further Contractibility

The residual type directly determines whether additional partial contractions remain possible: further contraction requires both a nonzero contravariant count and a nonzero covariant count in the residual type, since a contraction always needs one index of each variance.

Governing Transformation Rules

Each contravariant slot counted in the residual type contributes one factor of the basis-change Jacobian, and each covariant slot contributes one factor of its inverse, so the residual type fully determines the transformation law obeyed by the contracted tensor.

Special Cases

Purely Contravariant or Covariant Residual

If a contraction sequence exhausts all covariant indices while contravariant indices remain, the residual type takes the form $$(p-k,0)$$, describing a purely contravariant tensor that admits no further contraction until paired with a covariant index from another tensor.

Residual Type of Zero

When contraction reduces both counts to zero, the residual type $$(0,0)$$ corresponds to a scalar, marking the boundary case where partial contraction has become full contraction.

Residual Type Under Combined Tensors

When a partially contracted tensor is later combined with another tensor via an outer product, the residual type of the combination is obtained by adding the respective contravariant counts and covariant counts of each factor, before any further contraction is applied to the combined object.