12.11 Tensor Domain Restriction Operation

Tensor Domain Restriction Operation is the operation of confining a tensor's multilinear action to a chosen subspace of its original underlying vector space, producing a new tensor that agrees with the original wherever vectors or covectors are drawn exclusively from that subspace, while remaining otherwise undefined outside of it.

Formal Definition

Restricting the Vector Space Domain

For a tensor $$A$$ of type $$(p,q)$$ built over a vector space $$V$$, and given a subspace $$W\subseteq V$$, the domain restriction of $$A$$ to $$W$$ is the tensor $$A_{|}$$ defined by evaluating $$A$$ exclusively on vectors and covectors drawn from $$W$$ and its corresponding restricted dual:

for every choice of vectors $$w_k\in W$$, with the understanding that $$A_{|}$$ is only defined on this restricted set of arguments.

Preserving the Original Rule on the Smaller Domain

The restricted tensor does not introduce any new rule for computing outputs; it simply reuses the original tensor's multilinear formula, but only accepts arguments confined to the chosen subspace, leaving its behavior on vectors outside $$W$$ unspecified.

Componentwise Description

Restricting to a Basis of the Subspace

If a basis of $$W$$ is chosen, extended if necessary to a full basis of $$V$$, the components of the domain-restricted tensor are simply the components of the original tensor $$A$$ corresponding to index values that range only over the basis vectors of $$W$$, leaving out the components associated with directions outside the subspace.

A Subset of the Original Component Array

The restricted tensor's components form a subset of the array describing the original tensor, selected according to which basis vectors belong to the chosen subspace, rather than an entirely new independently computed array.

Why Domain Restriction Is Useful

Focusing on a Relevant Substructure

Domain restriction allows attention to be focused on how a tensor behaves when only a particular subspace of directions is relevant, which arises naturally when a physical or geometric problem confines its inputs to some meaningful lower-dimensional subset of the full space.

Simplifying Computations Within a Subspace

Working with the domain-restricted tensor can simplify computations when only vectors from the chosen subspace will ever be supplied as arguments, since the restricted tensor carries only the information relevant to that subspace.

Consequences for Type and Structure

The Restricted Tensor Has a Smaller Effective Vector Space

The domain-restricted tensor is naturally regarded as a tensor of the same type $$(p,q)$$, but now built over the smaller vector space $$W$$ instead of the original $$V$$, which affects its compatibility with other tensors during further operations such as addition, since those operations require matching underlying vector spaces.

Dependence on the Chosen Subspace

Different choices of subspace $$W$$ generally produce different restricted tensors, even from the same original tensor $$A$$, since the restriction depends entirely on which directions are retained.

Illustration