5.11.4 Tensor Product Scalar Distribution

Tensor Product Scalar Distribution is the identity, part of the tensor product's multilinearity, stating that a scalar multiplying any one factor of a tensor product can be pulled out and applied to the whole tensor, or equivalently moved to multiply any other single factor instead, so that the placement of a scalar among the factors of a tensor product carries no independent meaning of its own.

Statement of the Identity

For vectors v1, …, vn in their respective spaces and a scalar λ in the base field F, scalar distribution asserts

for every index i between 1 and n, meaning a scalar attached to any one factor can be extracted to scale the entire tensor, and by the same identity applied in reverse to a different index, the extracted scalar can then be reinserted into any other single factor instead.

Consequence: Freedom to Relocate a Scalar

Because the scalar can be pulled all the way out to the front and then reinserted into a different slot, scalar distribution directly yields the ability to move a scalar factor between any two positions in a tensor product without changing the resulting element.

The Two-Step Relocation Argument

To move a scalar from slot i to slot k, first extract it via (λvi) ⊗ ⋯ = λ(v1 ⊗ ⋯ ⊗ vi ⊗ ⋯ ⊗ vn), then reinsert it into slot k using the same identity read in the opposite direction, λ(v1 ⊗ ⋯ ⊗ vk ⊗ ⋯ ⊗ vn) = v1 ⊗ ⋯ ⊗ (λvk) ⊗ ⋯ ⊗ vn. Composing these two steps shows (λvi) ⊗ ⋯ ⊗ vk ⊗ ⋯ = v1 ⊗ ⋯ ⊗ (λvk) ⊗ ⋯, with all other factors unchanged, for any pair of indices i and k.

Source of Non-Uniqueness in Product Expressions

This relocation freedom is the direct origin of the well-known non-uniqueness of factor lists and product expressions for decomposable tensors: any decomposition of a scalar into a product of n scalars λ1 λ2 ⋯ λn = 1 can be distributed one factor per slot, producing infinitely many distinct-looking but equal product expressions for the same tensor, all related by scalar distribution applied repeatedly.

Distinction from Additive Distribution

Scalar distribution and additive distribution are the two separate halves of multilinearity in a given slot, and it is worth keeping them conceptually distinct even though both are consequences of the same underlying linearity.

Scalar Distribution Acts on a Single Term

Whereas additive distribution expands a tensor product of sums into many separate cross-terms, scalar distribution acts on a single decomposable term, only relocating a numerical multiplier among its factors without introducing any new terms or splitting anything into a sum.

Combined Use in General Linear Combinations

In practice, both identities are typically invoked together: expanding a linear combination of vectors in each slot first uses additive distribution to produce the individual cross-terms, and scalar distribution is then used within each resulting term to collect or relocate the numerical coefficients into a single position, commonly the front of the whole expression.

Well-Definedness Guaranteed by Construction

That scalar distribution holds is not something that must be separately checked once the tensor product has been constructed via its universal property or its quotient construction; it is guaranteed automatically by either construction.

From the Quotient Construction

In the free-vector-space quotient construction, the relations imposed to build the tensor product explicitly identify (λvi, …) with λ(v1, …, vi, …, vn) as formal symbols before passing to the quotient, so scalar distribution holds by definition of the quotient, not as a subsequently proved theorem.

From the Universal Property

Because the canonical map τ is required to be multilinear, and scalar compatibility in each slot is part of what multilinearity means, any construction satisfying the universal property automatically satisfies scalar distribution; this is one of the reasons different constructions of the tensor product, though built differently, are guaranteed to agree on this identity.

Illustrative Diagram

The chain of equalities traces a single scalar as it moves from the first factor, through the overall coefficient position, into the second factor, illustrating that its placement carries no independent significance.