12.11.4 Tensor Restriction Evaluation Preservation

Tensor Restriction Evaluation Preservation is the property guaranteeing that evaluating a domain-restricted tensor on arguments drawn from the target subspace produces exactly the same scalar or tensor result as evaluating the original, unrestricted tensor on those same arguments, ensuring that restriction changes only the tensor's available domain and not the numerical outcome of any evaluation still possible after restriction.

Statement of the Preservation Property

Matching Outputs on the Common Domain

For a tensor $$A$$ over a source domain $$V$$, restricted to a target subspace $$W$$ to produce $$A_{|W}$$, evaluation preservation asserts that for any vectors drawn from $$W$$:

holds identically, since the restricted tensor was defined precisely to reuse the original tensor's evaluation rule on this common domain, introducing no alteration to the computed value itself.

Immediate Consequence of the Definition

This preservation is not an additional fact requiring separate proof but follows directly from how domain restriction was defined in the first place: the restricted tensor's evaluation rule was declared to coincide exactly with the original tensor's rule whenever arguments come from the target subspace.

Why This Preservation Matters

Restriction Does Not Distort Existing Information

Evaluation preservation confirms that restricting a tensor's domain is a purely subtractive operation with respect to which arguments are permitted, rather than a modification of how the tensor behaves on the arguments it still accepts. Any calculation that only ever used vectors from $$W$$ would obtain identical results whether performed with the original tensor or its restriction.

Justifying the Use of Restricted Tensors as Substitutes

Because evaluation preservation holds, a restricted tensor can be safely used in place of the original tensor whenever a computation is guaranteed to involve only arguments from the target subspace, without any risk of altering the numerical outcome of that computation.

Interaction with Componentwise Structure

Consistency with the Componentwise Definition

Evaluation preservation is consistent with, and can be verified directly from, the componentwise description of restriction, since the components of the restricted tensor were defined to be exactly the corresponding components of the original tensor for index values lying within the target subspace's basis, guaranteeing that any contraction performed using these shared components yields the same scalar result.

Relationship to Domain Extension

A Converse Preservation Property

Domain extension exhibits an analogous preservation property in the opposite direction: an extended tensor evaluated on arguments from its original, smaller domain reproduces exactly the values of the original tensor before extension, mirroring the evaluation preservation seen in restriction and reflecting the complementary relationship between the two operations.

Illustration