13.13.4 Tensor Contraction Pair Index Compatibility

Tensor Contraction Pair Index Compatibility is the combined criterion applied during pair selection confirming that a candidate contravariant slot and a candidate covariant slot satisfy both the requirement of opposite variance and the requirement of matching dimension simultaneously, qualifying that specific pair as eligible for contraction. It names the joint suitability test performed on a proposed pairing before it is accepted as a valid selection, distinguishing this combined check from either requirement considered separately.

Conceptual Basis

Two Requirements Assessed Together

A candidate pair proposed during contraction pair selection must satisfy two distinct conditions at once: the two slots must carry opposite variance, one contravariant and one covariant, and they must range over spaces of the same dimension. Index compatibility refers to the joint satisfaction of both conditions by the specific pair under consideration.

Compatibility as a Gate on Selection

Before a candidate pair identified during selection can be accepted and passed on to actual summation, index compatibility functions as a gate, filtering out any proposed pairing that fails either the variance requirement, the dimension requirement, or both, leaving only pairs that satisfy the complete set of conditions.

Relation to the Broader Selection Process

While pair selection as a whole involves designating candidate contravariant and covariant slots from among those available on a tensor or tensors, index compatibility is the specific verification applied to a designated candidate pair, confirming that the designated choice is actually admissible before proceeding further.

Formal Description

Joint Statement of the Criterion

For a candidate contravariant slot with variance type upper and dimension $$m$$, and a candidate covariant slot with variance type lower and dimension $$n$$, index compatibility for this pair holds precisely when:

with both conditions required simultaneously for the pair to be treated as valid.

Applying the Criterion to a Multi-Index Tensor

For a tensor $$T_{kl}^{ij}$$, checking index compatibility for the candidate pair $$(i,k)$$ requires confirming both that $$i$$ is contravariant while $$k$$ is covariant, and that both range over the same dimension, before the contraction:

may be considered valid.

Failure Modes

A candidate pair fails index compatibility if either condition is violated: pairing two contravariant indices fails on variance grounds regardless of dimension, while pairing a contravariant and covariant index of differing dimension fails on dimensional grounds despite correct variance, and either failure alone is sufficient to disqualify the pair.

Properties

Necessity of Both Conditions

Index compatibility is not satisfied by meeting only one of its two constituent requirements; a pair with correct variance but mismatched dimension, or matched dimension but same-type variance, both fail the combined criterion equally.

Applicability Across All Pair Selections

Whether the candidate pair is drawn from a single tensor, as in trace contraction, or from two separate tensors, as in matrix multiplication or vector covector pairing, the same joint index compatibility criterion applies uniformly to the candidate pair under consideration.

Determinism Given a Fixed Candidate Pair

For any specific candidate pair proposed during selection, index compatibility is a fixed, determinate property, either holding or failing, with no ambiguity once the variance and dimension of both designated slots are known.

Practical Considerations

Standard Verification Step in Pair Selection

Confirming index compatibility for a candidate pair is a standard and necessary step immediately following the designation of contravariant and covariant slots during pair selection, ensuring that only admissible pairings proceed to the summation stage.

Guiding the Search Among Multiple Candidates

When several candidate pairs are possible on a tensor with multiple indices of each variance, checking index compatibility for each candidate in turn allows the full set of admissible pairs to be identified systematically, distinguishing valid options from those that must be excluded.

Role in Automated Tensor Systems

Computational systems that manipulate tensors symbolically or numerically typically implement index compatibility as an explicit validation check performed automatically whenever a contraction is requested, preventing the execution of operations on pairs that fail either the variance or dimension requirement.