10.20.3 Tensor Transformation Basis Pairing Check

Tensor Transformation Basis Pairing Check is the verification step confirming that the primary basis vectors and the dual basis vectors used throughout a tensor transformation remain correctly paired, satisfying their defining biorthogonality relation, both before and after the change of basis, so that covariant and contravariant components continue to combine into valid, well-defined contractions.

The Pairing Relation Being Checked

Biorthogonality of Basis and Dual Basis

The primary basis vectors $$e_i$$ and the dual basis vectors $${\epsilon }^j$$ must satisfy the defining pairing relation, evaluating to the Kronecker delta:

The basis pairing check confirms that this relation continues to hold after a change of basis is applied to both the primary and dual bases, since it is precisely this relation that gives the pairing between a covector and a vector its coordinate-independent meaning as a scalar.

Checking the Relation in the New Basis

After transforming both bases, the check verifies:

remains true, which follows automatically from the Jacobian product identity whenever the primary basis is transformed with the forward Jacobian and the dual basis is transformed with the inverse Jacobian, exactly as the tensorial rule prescribes.

Why This Pairing Might Break

Using the Wrong Matrix for One Basis

If, by error, both the primary basis and the dual basis were transformed using the same matrix rather than a matrix and its inverse respectively, the pairing relation would fail to reduce to the Kronecker delta after the transformation, since the required cancellation depends specifically on one basis using the forward Jacobian and the other using its inverse.

Inconsistent Choice of Dual Basis

A basis pairing check can also expose an error in which the dual basis chosen does not actually correspond to the primary basis being used, for instance if the dual basis was computed relative to a different, unrelated primary basis; in this case the pairing relation fails even before any coordinate transformation is applied, indicating that the error predates the transformation itself.

Practical Verification Procedure

Direct Computation of the Pairing

The most direct form of the check computes the pairing of every dual basis vector against every primary basis vector explicitly and confirms that the resulting array of numbers is exactly the identity matrix, with ones on the diagonal and zeros elsewhere, both before and after the transformation is applied.

Consistency With Component Contractions

Since a fully contracted scalar formed from a covector and a vector is built directly from this same pairing relation applied term by term, confirming that contracted scalars remain invariant under the transformation, as verified separately in the broader tensor transformation consistency check, provides an indirect but effective confirmation that the basis pairing has also been preserved correctly.

Diagram of the Pairing Check

Basis and Dual Basis Before and After

Higher-Rank Extensions of the Pairing Check

Tensor Product Basis Pairing

For a rank-two tensor built from a tensor product basis, the pairing check extends to verifying that the tensor product of dual basis vectors evaluated against the tensor product of primary basis vectors reproduces the expected product of Kronecker deltas, confirming the pairing structure is preserved not just for individual vectors and covectors but for the full tensor product basis used by higher-rank tensors.

Mixed Tensor Contraction Check

For a mixed tensor with both upper and lower indices, the basis pairing check can be extended to confirm that contracting the tensor against a matching combination of basis and dual basis vectors reproduces the correct individual tensor component, both before and after the transformation, offering a component-level verification that complements the purely structural pairing check on the bases alone.

Relationship to Other Consistency Checks

Foundational Role

The basis pairing check is, in a sense, the most fundamental of the various tensor transformation consistency checks, since the invariance of contracted scalars and the correctness of the tensorial rule itself both rely on this pairing relation continuing to hold; a failure detected by the contraction invariance check can often be traced back to an underlying failure of this more basic basis pairing relation.