13.14 Tensor Contraction Result Type

Tensor Contraction Result Type is the classification, by contravariant and covariant index counts, of the object produced once one or more contractions have been applied to a tensor or to a combination of tensors, encompassing the general rule by which this type is computed from the type of the inputs and the number of contractions performed. It identifies the general concept governing what kind of object, scalar or tensor of some reduced rank, emerges from any given contraction, serving as the umbrella notion beneath which specific outcomes such as full contraction, partial contraction, and the various named contraction cases are all particular instances.

Conceptual Basis

Type as the Central Descriptor of an Outcome

Every tensor is classified by its type, a pair of counts recording how many contravariant and how many covariant indices it carries. The contraction result type is this same classification applied specifically to the object obtained after a contraction, describing what has changed relative to the type of the tensors that entered the operation.

Determined by the Inputs and the Contraction Performed

The result type is not an independent property but is fully determined by the type of the original tensor or tensors together with the number and nature of the contractions applied, since each valid contraction removes exactly one contravariant and one covariant index from the overall count.

Encompassing Both Full and Partial Outcomes

The result type accounts for the entire spectrum of possible outcomes, from a tensor that retains many free indices after only a single contraction, to a scalar produced when every available index has been consumed, situating both extremes as points along a single continuum described by the same underlying rule.

Formal Description

General Formula for the Result Type

For a tensor of original type $$(p,q)$$ subjected to $$k$$ valid contractions, the contraction result type is:

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

Result Type When Combining Two Tensors

When two tensors of types $$(p_1,q_1)$$ and $$(p_2,q_2)$$ are combined by an outer product followed by $$k$$ contractions across the combined index set, the result type is:

Boundary Case Yielding a Scalar

The result type reaches its minimal form $$(0,0)$$ precisely when the number of contractions performed equals the smaller of the total contravariant and covariant counts available, marking the point at which the contraction result ceases to be a tensor and becomes a scalar.

Properties

Path Independence

The contraction result type depends only on the total number of contractions performed and the type of the original inputs, not on the specific order in which individual index pairs were contracted, provided each contraction respects the necessary variance and dimension requirements.

Conservation of the Type Difference

Across any sequence of contractions applied to a single tensor, the difference between the contravariant and covariant counts remains constant, since each contraction reduces both counts equally, making this difference an invariant quantity of the result type.

Determining Further Operability

The contraction result type directly determines what further operations remain possible: a nonzero result type permits additional contraction, multiplication by further tensors, or other tensorial operations, while a scalar result type admits only ordinary arithmetic.

Relationship to Named Contraction Cases

Full and Partial Contraction

Full contraction is the specific instance of the general rule in which the number of contractions performed exhausts the smaller of the two index counts, while partial contraction corresponds to any smaller number of contractions, both governed by the same underlying result type formula.

Trace, Matrix Multiplication, and Related Cases

The result types associated with trace contraction, matrix multiplication, inner products, and vector covector pairing each follow directly from applying the general contraction result type formula to tensors of the specific ranks characteristic of each of those named cases.

Practical Use in Predicting Outcomes

Before carrying out a proposed sequence of contractions, computing the anticipated contraction result type in advance allows the essential character of the final outcome, whether scalar or tensor of some given rank, to be known without performing the full computation.