13.7 Tensor Partial Contraction Operation

Tensor Partial Contraction Operation is the tensor algebra operation in which one or more, but not all, matched contravariant-covariant index pairs of a tensor are summed via the Einstein summation convention, reducing the tensor's rank by two for each pair contracted while leaving at least one contravariant or covariant slot free. It produces a tensor of lower rank than the original that still transforms nontrivially under a change of basis, distinguishing it from a full contraction, which consumes every index and yields a basis-independent scalar.

Conceptual Basis

Selective Index Pairing

Partial contraction begins from a tensor with a given set of contravariant and covariant indices and selects a proper subset of those indices to pair and sum, leaving the remaining indices untouched as free slots on the resulting object.

Rank Reduction Without Full Collapse

Because only a subset of index pairs is summed, the operation reduces rank incrementally rather than collapsing the tensor entirely, making it the mechanism by which higher-rank tensors are stepped down toward, but not necessarily all the way to, a scalar.

Relationship to the Trace Operation

When applied to a single mixed-index pair on an otherwise higher-rank tensor, partial contraction generalizes the familiar matrix trace, extracting a lower-rank object that summarizes the behavior of the tensor along the contracted directions while preserving information carried by the remaining indices.

Formal Description

Index Notation

For a tensor $$T_{kl}^{ij}$$, a partial contraction over the pair $$i,k$$ is written:

where the repeated index $$i$$ is summed over its full range, while $$j$$ and $$l$$ remain free.

Rank Change

For a tensor of type $$(p,q)$$, contracting $$k$$ pairs produces a tensor of type:

with the constraint $$k<\min (p,q)$$ required for the result to remain a nonscalar tensor.

Requirement of Matching Variance

Each contracted pair must consist of one contravariant and one covariant index ranging over the same dimension, since summation across mismatched variance or mismatched dimension is not defined within the standard tensor algebra framework.

Properties

Basis Dependence of the Result

Because free indices remain, the resulting tensor still carries transformation factors from the Jacobian of a coordinate change for each remaining contravariant slot and from its inverse for each remaining covariant slot, so the output is not invariant in the way a fully contracted scalar would be.

Order Independence Among Contracted Pairs

When multiple index pairs are contracted in a single partial contraction, the order in which the individual pairs are summed does not affect the final result, since summation over distinct dummy indices commutes.

Composability

A partially contracted tensor can serve as the input to further tensor operations, including additional contractions, products with other tensors, or symmetrization and antisymmetrization procedures applied to its remaining free indices.

Relationship to Other Contraction Types

Full Contraction

Full contraction is the limiting case of partial contraction in which every available index pair is summed, so that no free slots remain and the result is a scalar rather than a tensor.

Multiple Independent Partial Contractions

A single tensor may admit several different partial contractions depending on which index pairs are selected, each producing a distinct lower-rank tensor that emphasizes different structural relationships among the original indices.

Iterated Partial Contraction

Applying partial contraction repeatedly, contracting one pair at a time, is equivalent to applying a single partial contraction over the combined set of pairs, provided the pairs involved are disjoint and each satisfies the matching variance and dimension requirements.

Practical Role

Extracting Lower-Rank Structure

Partial contraction is the mechanism by which vectors, covectors, or lower-rank tensors are extracted from higher-rank tensors, such as obtaining a vector from a rank-three tensor by contracting one index against a metric or another auxiliary tensor.

Building Blocks for Invariant Quantities

Sequences of partial contractions are commonly used as intermediate steps toward a final full contraction, with each partial step reducing rank while preserving the specific directional information needed for subsequent operations.

Use in Multilinear Expressions

In expressions involving products of several tensors, partial contraction determines which indices from which factors are linked together, shaping the overall index structure of the resulting composite tensor before any further reduction is applied.