5.15.3 Tensor Product Formal Sum Relation

The Tensor Product Formal Sum Relation describes how tensor products combine vectors into multilinear forms through formal sums, foundational in algebraic structures.

Tensor Product Formal Sum Relation is the equivalence, holding within V ⊗ W, that identifies different finite formal sums of simple tensors whenever they represent the same underlying element after applying the additivity and homogeneity relations. It addresses the fact that a general tensor almost always has multiple, superficially different, representations as a sum Σ vᵢ ⊗ wᵢ, and it specifies precisely the conditions under which two such formal sums must be regarded as equal.

The General Formal Sum

Elements of the Tensor Product as Finite Sums

Every element t ∈ V ⊗ W can be written as a finite formal sum of simple tensors:

t = \sum_{i = 1}^{n} v_{i} \otimes w_{i}

since simple tensors span V ⊗ W, though such a sum is generally far from unique.

The Formal Sum Relation Statement

Two formal sums represent the same element of V ⊗ W:

\sum_{i = 1}^{n} v_{i} \otimes w_{i} = \sum_{j = 1}^{m} v_{j}' \otimes w_{j}'

precisely when their difference can be reduced to zero using only the additivity and homogeneity relations, equivalently, when the corresponding elements of the free module F(V × W) differ by an element of the relation submodule R.

Sources of Non-Uniqueness in Formal Sums

Rebalancing Scalars Between Factors

A formal sum can be rewritten by shifting scalars between the two factors of any term, using homogeneity: (cv) ⊗ w = v ⊗ (cw), without changing the total sum.

Merging Terms with a Shared Factor

Terms sharing a common right-hand factor can be merged via additivity:

u \otimes w + v \otimes w = (u + v) \otimes w

reducing the number of terms in the formal sum without changing the element it represents.

Basis Expansion Producing Canonical Formal Sums

If {eᵢ} and {fⱼ} are bases of V and W, every formal sum can be rewritten, using additivity and homogeneity repeatedly, into the canonical form:

\sum_{i, j} c_{i j} (e_{i} \otimes f_{j})

in which the coefficients cᵢⱼ are uniquely determined, giving a canonical representative for every equivalence class of formal sums.

Diagram of Equivalent Formal Sums

Formal Verification of Equality

Reduction to the Coefficient Matrix

Because the canonical basis-expansion form is unique, verifying whether two formal sums represent the same tensor reduces to expanding both in a common basis and comparing the resulting coefficient matrices (cᵢⱼ) entry by entry, providing a fully algorithmic test for equality of tensors given as formal sums.

Rank as an Invariant of the Formal Sum

While the specific terms in a formal sum are not unique, the minimal number of terms needed, which equals the rank of the coefficient matrix, is an invariant of the tensor itself and does not depend on which particular formal sum representation was used to compute it.

Why the Relation Matters

Ensuring Operations Are Well-Defined

Any operation on V ⊗ W defined by specifying its effect on formal sums, such as a linear map given by f(Σ vᵢ ⊗ wᵢ) = Σ g(vᵢ, wᵢ) for some bilinear g, must be checked against the formal sum relation to confirm the result does not depend on which representation of the input was chosen; this is exactly the well-definedness check guaranteed by the universal property when g is genuinely bilinear.

Practical Simplification in Symbolic Computation

In symbolic and computational treatments of tensors, the formal sum relation is what justifies simplification routines that combine like terms, factor out scalars, or reduce a tensor expression to a minimal number of simple-tensor terms, all without altering the value of the underlying tensor.