5.7.1 Tensor Simple Product Form

Tensor Simple Product Form is a foundational concept in tensor algebra, representing tensors as products of simpler components within a structured mathematical framework.

Tensor Simple Product Form is the name given to the notation v1 ⊗ v2 ⊗ ⋯ ⊗ vn used to write a simple (elementary, rank-one) tensor as an explicit chain of tensor-product symbols connecting one vector from each participating factor space, as opposed to writing the same element as an expanded array of numerical coordinates or as a sum of several such chains.

Formal Definition

For vector spaces V1, V2, …, Vn over a field F, an element t of the tensor product V1 ⊗ V2 ⊗ ⋯ ⊗ Vn is said to be given in simple product form when it is written as

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

with each vi in Vi. The word "simple" here marks the tensor as an elementary term rather than a sum, and the word "product" marks the notation as an explicit chain of tensoring operations rather than an expanded coordinate list. A tensor admits a simple product form exactly when it has tensor rank at most one.

Elementary Terms Versus General Expressions

Simple product form is the building block out of which the general expression for any tensor is assembled, since the tensor product space is spanned by exactly these elementary terms.

The General Sum-of-Simples Expression

Any element of the tensor product space can be written as a finite sum

t = \sum_{k = 1}^{r} v_{1}^{(k)} \otimes \dots \otimes v_{n}^{(k)}

of terms, each individually in simple product form. A tensor is in simple product form itself precisely when r can be taken equal to one, that is, when a single elementary term already equals the whole tensor.

Distinguishing "Simple" from "Sum" Notation

Because both a single elementary term and a sum of many elementary terms are built from the same tensor-product symbol, care is needed to distinguish an expression genuinely in simple product form from a sum that merely contains several such expressions added together; only the former names a rank-one tensor, while the latter, in general, does not reduce back to any single simple product form.

Notational Conventions

Several conventions govern how simple product form is written across different areas of mathematics and its applications.

Order of Factors

The order of the factors in v1 ⊗ v2 ⊗ ⋯ ⊗ vn is tied to the order of the spaces V1, V2, …, Vn in the tensor product; permuting the written order of the factors without also permuting the spaces changes which space each written vector is understood to inhabit, so the ordering is a load-bearing part of the notation rather than a stylistic choice.

Parenthesization and Associativity

Because the tensor product is associative up to a canonical isomorphism, the simple product form v1 ⊗ (v2 ⊗ v3) and (v1 ⊗ v2) ⊗ v3 are typically identified and written without parentheses as v1 ⊗ v2 ⊗ v3, treating the tensor product of three or more factors as a single multilinear operation rather than a repeated binary one.

Common Alternative Terms

The same notion appears in the literature under the names elementary tensor, decomposable tensor, rank-one tensor, and pure tensor, all referring to an element expressible in simple product form; the choice of term often reflects the subfield — "elementary" and "simple" in algebra, "rank-one" in numerical multilinear algebra, and "pure" and "decomposable" in more geometric or representation-theoretic treatments.

Simple Product Form and Multilinear Maps

Simple product form is the form in which the universal property of the tensor product is naturally invoked, since multilinear maps act directly on the factors appearing in this notation.

Evaluation Rule

If φ is a multilinear map on V1 × V2 × ⋯ × Vn with linear extension φ̂ on the tensor product, then

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

This equation applies directly to any tensor written in simple product form, and, by linearity, extends termwise to any sum of tensors each individually presented in simple product form.

Simple Product Form as a Computational Shortcut

Many algorithms deliberately keep tensors in simple product form, or as short sums of such forms, rather than expanding to full coordinate arrays, because operations like evaluating a multilinear map, computing an inner product, or applying a contraction can be carried out factor by factor on the simple product form at a cost proportional to the sum of the factor dimensions rather than their product.

Simple Product Form in Coordinates

When bases are fixed for each Vi, expanding a simple product form into coordinates produces a tensor whose entries are products of the corresponding coordinates of the individual factors, a special structural pattern not shared by tensors of higher rank.

t_{i_{1} \dots i_{n}} = {(v_{1})}_{i_{1}} {(v_{2})}_{i_{2}} \dots {(v_{n})}_{i_{n}}

This entrywise product structure is what a coordinate array must satisfy for it to correspond to a tensor in simple product form, and it is the basis for the rank-one tests used to recognize such tensors from their coordinates.

Illustrative Diagram

The unbroken chain of tensor-product symbols in the diagram is precisely what marks the expression as being in simple product form, as opposed to a sum of several such chains.