13.20.2 Tensor Contraction Pair Validity Check

Ensuring valid tensor contraction pairs is essential for accurate calculations in tensor algebra, verifying index matching and dimension consistency.

Tensor Contraction Pair Validity Check is the verification test confirming that a proposed contraction pairs exactly one contravariant index with exactly one covariant index, rejecting any proposed pairing between two indices of the same variance, since contraction is defined only through the natural pairing between a vector space and its dual.

Definition

For a proposed contraction of index $i$ from tensor $A$ against index $j$ from tensor $B$ , the pair validity check requires:

variance (i) \neq variance (j)

that is, one of the two indices must be contravariant, or upper, while the other must be covariant, or lower. A pairing of two upper indices, or of two lower indices, fails this check.

Why Variance Mismatch Is Required

Natural Pairing Requirement

Contraction is mathematically grounded in the canonical bilinear pairing between a vector space $V$ and its dual space $V^{, defined by evaluation of a covector on a vector. This pairing exists specifically because one space is contravariant and the other covariant; no analogous canonical pairing exists between two copies of V alone, or between two copies of V^{} alone, without introducing additional structure such as a metric. Consequence of Ignoring the Check Attempting to sum over two indices of the same variance without a metric produces an expression that does not transform as a tensor under a change of basis, since the transformation laws for two contravariant indices do not cancel in the way required for invariance, unlike the transformation laws for one contravariant and one covariant index. Relation to Metric-Assisted Contraction Same-Variance Pairing With a Metric A pairing between two indices of the same variance can be made valid by first applying metric simplification to raise or lower one of the two indices, converting the pairing into a valid upper-lower pair before the contraction itself is checked and performed. A^{i} B^{j} g_{i j} is valid: the metric supplies the needed lower indices Distinction From Plain Contraction The pair validity check applies to the contraction step itself; if a metric factor is present in the expression, the check is applied only after the metric has been used to adjust variance, not to the raw same-variance indices in isolation. Diagram Valid: upper paired with lower A B Invalid: upper paired with upper Placement Within Verification The pair validity check is applied alongside the dimension check and the index balance check as one of the foundational structural tests within the tensor contraction verification procedure, ensuring that a proposed contraction is not merely numerically well defined but also mathematically meaningful as a genuine tensor operation before any further simplification or evaluation proceeds. Related content 13.20.1 Tensor Contraction Index Balance Check 13.20.3 Tensor Contraction Dimension Check 13.20.4 Tensor Contraction Result Type Check 13.20.5 Tensor Contraction Invariance Check Location Codartium Formal Sciences Mathematics Algebra Tensor Algebras 13 Tensor Contractions 13.20 Tensor Contraction Verification Procedure 13.20.2 Tensor Contraction Pair Validity Check (You are) © 2025 Codartium — Made for everyone, free. About Contact Disclaimer Privacy Terms}$