12.12 Tensor Domain Extension Operation

Tensor Domain Extension Operation is the operation of taking a tensor originally defined over a smaller vector space and producing a new tensor defined over a larger vector space containing the original as a subspace, in a manner that agrees with the original tensor whenever its arguments happen to come from that original subspace.

Formal Definition

Extending to a Larger Vector Space

For a tensor $$A$$ of type $$(p,q)$$ defined over a vector space $$W$$, and given a larger vector space $$V$$ containing $$W$$ as a subspace, domain extension produces a tensor $$A^{~}$$ defined over the whole of $$V$$ that agrees with $$A$$ whenever all supplied arguments lie in $$W$$:

for every choice of arguments $$w_k\in W$$, while $$A^{~}$$ is additionally defined, according to some specified extension rule, on arguments drawn from the larger space $$V$$.

The Simplest Extension Rule: Extension by Zero

A common and natural choice for the extension rule assigns a value of zero whenever any supplied argument lies in a chosen complementary subspace outside of $$W$$, so that the extended tensor's action is fully determined once a complement to $$W$$ within $$V$$ has been selected.

Componentwise Description

Filling in New Components

Choosing a basis of $$V$$ that extends a basis of $$W$$, the components of $$A^{~}$$ corresponding to index values entirely within the original basis of $$W$$ simply copy the components of $$A$$, while every new component involving at least one index value from outside $$W$$ is assigned according to the chosen extension rule, commonly set to zero.

Why Domain Extension Is Useful

Embedding a Smaller Tensor Into a Larger Framework

Domain extension allows a tensor originally understood only in relation to a limited subspace to be incorporated into computations involving the full, larger vector space, without discarding any of the information already captured by the original tensor on its native subspace.

Complementary Relationship to Restriction

Domain extension and domain restriction are complementary operations: restricting an extended tensor back to the original subspace $$W$$ recovers exactly the original tensor $$A$$, reflecting the requirement that the extension agree with the original tensor on its native domain.

Non-Uniqueness of the Extension

Dependence on the Chosen Extension Rule

Unlike restriction, which is uniquely determined once the subspace is chosen, domain extension is generally not unique, since many different extension rules could agree with the original tensor on $$W$$ while differing arbitrarily on arguments outside $$W$$. A specific extension rule, such as extension by zero, must be explicitly specified for the operation to produce a definite result.

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