5.6.3 Tensor Pure Element Product Form

Tensor Pure Element Product Form is the representation of a pure (simple, decomposable) element of a tensor product space as an explicit chain of tensor-product symbols joining its factor vectors, written v1 ⊗ v2 ⊗ ⋯ ⊗ vn rather than as an abstract array of coefficients. The product form is the syntactic expression of an element, distinguishing how it is written from the underlying factor list, which is the sequence of vectors that populate the expression.

Formal Definition

Given vector spaces V1, V2, …, Vn over a field F, the product form of an element t in V1 ⊗ V2 ⊗ ⋯ ⊗ Vn is the expression

where each vi belongs to Vi. An element that admits such an expression is called pure; the product form is precisely this admitted expression, taken as notation rather than as a computed value. Two product forms are considered the same expression only if they agree symbol for symbol, whereas the elements they denote may coincide even when the written forms differ, since tensor multiplication is multilinear and not injective on tuples.

Product Form Versus Coordinate Form

Every element of a finite-dimensional tensor product space can also be written in coordinate form, as a sum of scalar multiples of basis tensors indexed by an n-tuple of basis indices. The product form and the coordinate form are two different ways of naming the same mathematical object when the element happens to be pure.

Expansion from Product Form to Coordinates

If each factor vi is written in a chosen basis as a linear combination of basis vectors, distributing the tensor product over these sums converts the product form into coordinate form. This distribution uses multilinearity in each slot independently: the tensor product of a sum in one factor equals the sum of the tensor products obtained by tensoring each term in that sum with the other factors held fixed.

Collapse from Coordinates to Product Form

The reverse direction — recognizing that a tensor given in coordinate form is secretly expressible as a single product form — is not always possible. A coordinate array corresponds to a product form exactly when its associated multi-way array has tensor rank one, which for matrices (the two-factor case) is equivalent to every 2-by-2 minor vanishing, and which for higher-order tensors requires checking rank-one conditions across all unfoldings of the array.

Non-Uniqueness of Product Form

Although the product form is a specific written expression, more than one product form can denote the identical element, because scalar factors can be redistributed among the tensored vectors without changing their product.

Scalar Rebalancing

for any scalar λ, illustrating that the scalar attached to any one factor can migrate freely to any other factor without altering the value of the whole expression. Consequently the product form should be understood as representing an equivalence class of written expressions rather than a single canonical string, unless a normalization rule is imposed.

Ordering Conventions

The order in which the factors are written matters, since the factor spaces V1, …, Vn are generally distinct and the tensor product is not commutative across different spaces in the way ordinary multiplication is commutative within a single field. Reordering the written factors changes which space each position is understood to belong to, so a change of order is only valid when accompanied by a corresponding relabeling of the underlying spaces, such as through an explicit isomorphism that swaps two tensor factors.

Role in Multilinear Algebra

The product form is the natural notation in which the defining property of the tensor product — its universal property with respect to multilinear maps — is stated and used.

Universal Property Restated in Product Form

For any multilinear map φ from the Cartesian product V1 × V2 × ⋯ × Vn into a target space W, there is a unique linear map φ̂ from the tensor product into W satisfying

on every element written in product form. This identity is stated exclusively for elements in product form because it is only for such elements that the right-hand side, an evaluation of φ on an explicit tuple, is directly meaningful; general elements must first be written as sums of product forms before the identity can be applied termwise.

Spanning by Product Forms

Even though not every element of the tensor product space is itself expressible in a single product form, the collection of all elements that are so expressible spans the entire space: every element is a finite sum of elements in product form. This spanning property is part of the standard construction of the tensor product and explains why product forms, despite describing only a subset of elements individually, are sufficient to generate the whole space through linear combination.

Product Form in Tensor Networks and Rank Decomposition

In applied and computational settings, writing tensors as sums of terms in product form is the basis of rank decomposition methods.

Rank Decomposition

A general tensor t is written as a sum of r terms, each in product form:

The smallest such r for which this decomposition exists is the tensor rank of t, and each of the r summands is itself a product form contributed by one rank-one term. Algorithms for computing or approximating such decompositions, including alternating least squares methods, operate directly on the factor vectors that make up each product-form term.

Diagrammatic Notation

In tensor network diagrams, a product-form term is typically drawn as a set of disconnected nodes, one per factor, since no index is shared between the factors of a single pure term; shared indices and internal connections appear only once multiple product-form terms are combined or contracted together.

Illustrative Diagram

Each circle in the diagram is one factor of the product form; the absence of any line joining the circles reflects that a pure tensor written in product form carries no internal contraction, only the ordered juxtaposition of independent factors.