5.8.1 Tensor Decomposable Product Expression

Tensor Decomposable Product Expression represents multi-linear relationships through structured tensor product decompositions in algebra.

Tensor Decomposable Product Expression is the explicit written form v1 ⊗ v2 ⊗ ⋯ ⊗ vn that exhibits a decomposable tensor's factorization directly in the notation, functioning as the certificate that proves an element is decomposable by displaying its factors rather than merely asserting their existence.

Formal Description

For vector spaces V1, …, Vn over a field F and their tensor product T, a decomposable product expression for an element t of T is a specific tuple (v1, …, vn) with vi in Vi, together with the notational claim

t = v_{1} \otimes v_{2} \otimes \dots \otimes v_{n}

An element of T is decomposable exactly when at least one such expression exists for it; the expression itself is the concrete data witnessing that fact, as opposed to the abstract property of decomposability, which only asserts existence.

Expression Versus Property

It is useful to separate the property of being decomposable from any particular product expression witnessing it, since the two carry different logical weight.

Existence Versus Exhibition

Decomposability is an existential statement: there exist vectors whose product equals t. A decomposable product expression is the exhibition of one such witness. Two different mathematicians could agree that a given tensor is decomposable while writing down different, equally valid product expressions for it, since the witnessing tuple is not unique.

Verifying an Expression

Given a candidate expression (v1, …, vn), verifying that it correctly witnesses decomposability of a target tensor t reduces to expanding the tensor product of the candidate factors in coordinates and checking equality with the coordinates of t — a direct, mechanical computation requiring no search, in contrast to the harder problem of finding such an expression from scratch.

Constructing Product Expressions

Several standard routes produce a decomposable product expression once decomposability of a target tensor has been established or is assumed from context.

From a Rank-One Matrix

In the two-factor case, if a matrix M is known to have rank one, a product expression is obtained from any nonzero column c of M (up to scale, one factor) and the row of coefficients expressing every column of M as a multiple of c (the other factor, up to compensating scale).

From a Singular Value Decomposition

Taking the singular value decomposition of a rank-one matrix produces a normalized product expression M = σ u vᵀ directly, with u and v unit vectors and σ a nonnegative scalar absorbing the overall magnitude, giving one particular, canonically chosen representative among the many valid product expressions for the same tensor.

From Symmetric or Exterior Structure

In the exterior algebra, a decomposable p-vector's product expression — a wedge product w1 ∧ ⋯ ∧ wp — can be constructed from any basis of the p-dimensional subspace the p-vector represents; different bases of the same subspace give different, but equally valid, product expressions for the same decomposable element, differing only by an overall nonzero scalar (the determinant of the basis change).

Non-Uniqueness and Equivalence Classes of Expressions

Because scalar weight can migrate between factors without changing their product, a decomposable tensor is associated not to one product expression but to an entire equivalence class of them.

The Equivalence Relation

Two product expressions (v1, …, vn) and (v1′, …, vn′) denote the same tensor exactly when there exist nonzero scalars λ1, …, λn with

v_{i}' = λ_{i} v_{i} for every i, and \prod_{i = 1}^{n} λ_{i} = 1

Every decomposable tensor corresponds to exactly one such equivalence class of product expressions, so a canonical or normalized expression can be selected by fixing a representative within the class, for instance by requiring every factor except the last to be a unit vector.

Practical Impact

Because of this non-uniqueness, algorithms that produce a decomposable product expression as output — for example, from a rank-one approximation routine — should be understood as returning one representative of the equivalence class, not the unique answer, and downstream comparisons between two computed expressions for what should be the same tensor must account for this scalar redistribution before concluding the results disagree.

Role in Multilinear Map Evaluation

A decomposable product expression is precisely the form required to invoke the universal property of the tensor product directly, since multilinear maps take tuples of vectors as their natural input.

\hat{φ} (v_{1} \otimes \dots \otimes v_{n}) = φ (v_{1}, \dots, v_{n})

Having an explicit product expression in hand converts the abstract linear map φ̂ on the tensor product into a directly computable evaluation of the underlying multilinear map φ on the factor tuple, which is the entire practical motivation for constructing such expressions rather than working with tensors only in coordinate form.

Illustrative Diagram

The arrow marks the passage from the abstract claim that t is decomposable to the concrete product expression that exhibits and proves the claim by displaying the factors themselves.