5.11.5 Tensor Product Additive Distribution

The Tensor Product Additive Distribution describes how tensor products distribute over addition in algebraic structures, forming a foundational property in tensor algebra.

Tensor Product Additive Distribution is the full, simultaneous expansion of a tensor product whose factors are each themselves given as sums, producing a single grand sum with one term for every combination of summands chosen independently from each factor, directly generalizing the elementary distributive law of ordinary multiplication to the multilinear setting of the tensor product.

Statement of the Distribution

For vectors expressed as sums in each factor space,

v_{1} = \sum_{i} a_{1, i}, …, v_{n} = \sum_{j} a_{n, j}

the full additive distribution gives

v_{1} \otimes \dots \otimes v_{n} = \sum_{i, \dots, j} a_{1, i} \otimes \dots \otimes a_{n, j}

with a term for every combination obtained by picking one summand from each of the n sums independently, and no cross-terms omitted.

Derivation from Single-Slot Linearity

Additive distribution is not a separate axiom; it is obtained by applying the additivity identity in one slot at a time, iterated across every factor.

Iterating Additivity Slot by Slot

Expanding first the sum in v1 using first-slot additivity produces a sum of terms of the form a1,i ⊗ v2 ⊗ ⋯ ⊗ vn; expanding the sum in v2 within each of these terms using second-slot additivity further splits each into terms a1,i ⊗ a2,j ⊗ v3 ⊗ ⋯ ⊗ vn; repeating this process through all n slots produces the full grand sum, with the total number of applications of additivity equal to n − 1 rounds of expansion (one fewer than the number of factors, since the first factor's expansion starts the process).

Order-Independence of the Expansion

Because the order in which the n slots are expanded does not affect the final collection of cross-terms — only the intermediate bookkeeping — additive distribution produces the same grand sum regardless of which factor is expanded first, a consequence of the individual slot-linearity identities being independent of one another.

Combinatorial Growth of Cross-Terms

If the i-th sum has mi terms, the fully distributed expression contains m1 · m2 ⋯ mn cross-terms, a multiplicative blow-up directly analogous to, and structurally identical with, the multiplicative growth of tensor product dimension itself.

Term Count Versus Basis Expansion

When each vi is itself expanded in a chosen basis (so that mi = di, the dimension of Vi), additive distribution reproduces exactly the basis expansion of the decomposable tensor v1 ⊗ ⋯ ⊗ vn, with the coefficient of each basis tensor equal to the product of the corresponding factor coefficients; additive distribution is thus the mechanism, not merely an analogy, behind the entrywise multiplicative coordinate pattern of decomposable tensors.

Practical Consequence for Symbolic Computation

Any symbolic or hand computation that expands a tensor product of sums must account for this multiplicative term count, since omitting even one combination of summands produces an incorrect, incomplete expansion; software implementing tensor algebra symbolically typically automates this expansion precisely because manual bookkeeping of all cross-terms becomes error-prone once more than two or three summands per factor are involved.

Distribution Combined with Sums of Decomposable Tensors

Additive distribution also governs how a tensor already expressed as a sum of several decomposable terms interacts with further tensoring by an additional factor.

Distributing an Extra Factor Across an Existing Sum

If t = Σk (decomposable term k) is a tensor in V1 ⊗ ⋯ ⊗ Vn already given as a sum of r decomposable terms, then t ⊗ w for a vector w in a further factor space distributes as Σk (decomposable term k) ⊗ w, again by a single application of additivity in the newly introduced slot, showing that additive distribution applies uniformly whether the sums being distributed originate from expanding a single vector or from an existing rank decomposition.

Illustrative Diagram

Each cell of the grid is one of the six cross-terms produced by fully distributing a two-term sum against a three-term sum, illustrating the multiplicative term count of additive distribution.