13.20.5 Tensor Contraction Invariance Check

Tensor Contraction Invariance Check is the verification test confirming that a contracted result transforms correctly as a genuine tensor of its predicted type under a change of basis, distinguishing an object that merely looks like a valid contraction in one coordinate system from one that truly satisfies the transformation law required of a tensor in every admissible coordinate system.

Definition

For a computed result $$R$$ claimed to be a tensor of type $$(p,q)$$, the invariance check requires that under an arbitrary change of basis described by a transformation matrix $$\Lambda$$, the components of $$R$$ transform according to the standard rule:

with one factor of $$\Lambda$$ or its inverse appearing for each upper or lower index respectively, and no extra or missing factors.

Why the Check Is Needed

Contraction Guarantees Invariance in Principle

A properly performed contraction, pairing exactly one upper index with one lower index as required by the pair validity check, is guaranteed by the general theory of tensors to produce an object that transforms correctly; the invariance check exists to catch cases where an error elsewhere in the computation, such as an unnoticed same-variance pairing or a miscounted index, has produced an object that fails this guarantee in practice.

Distinguishing Coincidental Agreement

A computed array of numbers might happen to satisfy a desired equation in one specific coordinate system purely by chance or by an error that cancels out numerically in that particular basis; the invariance check, by testing behavior under an actual change of basis, exposes such coincidental agreement as distinct from genuine tensorial correctness.

Procedure

Step One: Choose a Nontrivial Basis Change

Select a transformation matrix $$\Lambda$$ that is not the identity, ensuring the test is capable of detecting a genuine transformation failure.

Step Two: Recompute in the New Basis

Independently recompute the original contraction entirely within the new basis, starting from the transformed input tensors, to obtain $$\tilde{R}$$ directly.

Step Three: Compare to the Transformation Law

Apply the standard transformation law to the original result $$R$$ to predict what $$\tilde{R}$$ should be, and compare this prediction against the independently recomputed value from step two.

Diagram

Placement Within Verification

The invariance check is the deepest and most conceptually demanding test within the tensor contraction verification procedure, extending beyond the structural checks of dimension, index balance, and pair validity, and beyond the outcome check of result type, to confirm the fundamental defining property of tensors themselves: consistent, predictable behavior under any admissible change of coordinate system.