13.13 Tensor Contraction Pair Selection

Tensor Contraction Pair Selection is the overall process of designating which specific contravariant index and which specific covariant index, drawn from one tensor or from two separate tensors, are to be identified and summed together in a given contraction, encompassing the choice of both the contravariant slot and the covariant slot as a single coordinated decision. It names the deliberative step that precedes the mechanical act of summation, distinguishing the question of which indices to contract from the question of how the resulting summation is carried out once the pair has been fixed.

Conceptual Basis

Selection as a Precondition for Contraction

Before any contraction can be performed, the specific indices to be paired must be identified, since tensors of rank greater than one may carry several indices of each variance, and the summation rule alone does not specify which particular upper and lower indices are meant to be linked in any given operation.

Two-Sided Nature of the Decision

Because a valid contraction always requires exactly one contravariant index and one covariant index, pair selection is inherently a two-sided decision, requiring the simultaneous or sequential designation of one slot from each variance category before the pairing is complete.

Distinctness From the Contraction Operation Itself

Pair selection concerns only the identification of which indices participate, not the arithmetic of summing over them; once a pair has been selected, the actual contraction proceeds according to the covariant contravariant summation rule, applying uniformly regardless of which specific pair was chosen.

Formal Description

Enumerating Candidate Pairs

For a tensor with $$p$$ contravariant indices and $$q$$ covariant indices of compatible dimension, pair selection amounts to choosing one of the $$p\times q$$ possible combinations of a contravariant slot and a covariant slot to be identified as a single contracted index.

Notating a Selected Pair

Given a tensor $$T_{kl}^{ij}$$, selecting the pair consisting of $$i$$ and $$k$$ results in:

as distinct from selecting the pair consisting of $$j$$ and $$k$$, which would instead produce a summation over a different index combination.

Selection Across Two Tensors

When the contravariant and covariant slots belong to separate tensors, as in matrix multiplication or vector covector pairing, pair selection additionally specifies which tensor contributes each half of the pair, since this determines the overall structure of the resulting composed or contracted object.

Properties

Non-Uniqueness in General

For tensors carrying more than one index of either variance, pair selection is generally non-unique, meaning several distinct valid selections exist, each of which constitutes a legitimate contraction yielding, in most cases, a different resulting tensor.

Dependence of the Result on the Selection Made

Because different selected pairs sum over different directional information within the original tensor or tensors, the outcome of a contraction depends critically on which pair selection has been made, making explicit specification of the selection essential whenever ambiguity is possible.

Compatibility as a Constraint on Selection

Not every combination of a contravariant slot and a covariant slot constitutes a valid selection, since the two chosen slots must range over compatible dimensions, restricting pair selection to those combinations satisfying this dimensional requirement.

Practical Considerations

Necessity in Multi-Index Tensors

While a tensor with only one contravariant and one covariant index requires no explicit selection, since only one candidate pair exists, any tensor with additional indices demands that pair selection be stated explicitly to avoid ambiguity about which contraction is intended.

Role in Symbolic Computation Systems

Symbolic tensor algebra systems must represent pair selection explicitly as part of any contraction operation, tracking precisely which contravariant and covariant slots have been designated so that the correct summation is applied when the expression is evaluated.

Foundation for Naming Specific Contraction Cases

The various named contraction cases, including trace contraction, matrix multiplication, inner products, and vector covector pairing, each correspond to a particular pattern of pair selection applied to tensors of specific rank, situating pair selection as the underlying process from which these more specific cases are derived.