5.16.5 Tensor Product Formal Sum Reduction

Tensor Product Formal Sum Reduction is the systematic process of rewriting a formal sum of simple tensors into an equivalent sum with fewer terms, or into a canonical basis-expansion form, by repeatedly applying the additivity and homogeneity relations until no further simplification is possible. This reduction process is the practical, algorithmic counterpart to the abstract equivalence relation defining V ⊗ W: it is how one actually computes a minimal or standard representative for a given tensor rather than merely knowing, in principle, that such representatives exist.

The Basic Reduction Moves

Merging Terms with a Shared Factor

Whenever two terms in a formal sum share an identical factor on one side, they can be merged using additivity:

reducing the term count by one each time such a merge is applied.

Absorbing Scalars via Homogeneity

A scalar coefficient sitting outside a simple tensor can be absorbed into either factor:

allowing every term to be normalized into the pure product form v' ⊗ w' without a separate leading coefficient.

Eliminating Zero Terms

If a term evaluates to 0 ⊗ w or v ⊗ 0, it can be dropped from the sum entirely, since both expressions equal the zero tensor by the consequence of bilinearity noted earlier.

Reduction Toward a Canonical Basis Form

Full Expansion in Fixed Bases

Given bases {eᵢ} and {fⱼ}, any formal sum can be reduced completely by first expanding every vᵢ and wᵢ in coordinates, distributing via additivity to obtain a sum purely of basis tensors eᵢ ⊗ fⱼ, and then merging all terms sharing the same basis tensor by summing their coefficients.

Termination of the Reduction Process

This expansion-and-merge procedure always terminates in a finite number of steps, since the number of distinct basis tensors eᵢ ⊗ fⱼ available (when V and W are finite-dimensional) is finite, bounding the number of merge operations needed.

Uniqueness of the Reduced Form

The fully reduced form obtained this way, Σ cᵢⱼ (eᵢ ⊗ fⱼ) with each basis tensor appearing at most once, is unique for a given tensor, since {eᵢ ⊗ fⱼ} is a basis and coefficients relative to a basis are uniquely determined.

Reduction Toward Minimal Rank, Not Just Canonical Form

Distinguishing Basis Reduction from Rank Reduction

Reducing to the canonical basis form is not the same as finding the minimal-length representation of a tensor: the canonical form may have many nonzero coefficients cᵢⱼ even when the tensor has a much smaller rank, since rank-reduction requires finding an entirely different, non-basis, set of vectors achieving the minimum term count.

Rank Reduction via Matrix Techniques

Reducing a tensor to a minimal-rank formal sum is computed, once the canonical coefficient matrix (cᵢⱼ) is known, by applying standard matrix decomposition techniques, such as row reduction or singular value decomposition, to express the matrix as a sum of the fewest possible rank-one matrices, each corresponding to a single simple tensor term.

Worked Example of Reduction

Starting Formal Sum

Consider the formal sum:

Reducing by Expansion and Merging

Expanding the second term by additivity gives e₁⊗f₁ + e₂⊗f₁; combining with the first term yields 2(e₁⊗f₁) + e₁⊗f₁ + e₂⊗f₁, which merges to the reduced form:

Visualization of the Reduction Process

Significance of Reduction as a Procedure

Practical Necessity for Symbolic Computation

Formal sum reduction is the concrete algorithm implemented in computer algebra systems whenever tensors are manipulated symbolically, ensuring that expressions built from many intermediate operations do not grow unboundedly in term count and remain comparable for equality testing.

Reduction as Confirmation of the Equivalence Relation

Every successful reduction from one formal sum to another is, at the level of the free module, a demonstration that the two sums differ by an element of the relation submodule R, making formal sum reduction the operational, step-by-step realization of the abstract equivalence relation defining the tensor product.