5.6.2 Tensor Pure Element Factor List

Explore how tensor pure element factors are systematically listed and their role in tensor algebra structures.

Tensor Pure Element Factor List is the ordered tuple of vectors, drawn one from each of the vector spaces participating in a tensor product, whose tensor product produces a given pure (also called simple or decomposable) element of that tensor product space. For a pure element written as v1 ⊗ v2 ⊗ ⋯ ⊗ vn, the factor list is the sequence (v1, v2, …, vn) itself, treated as the canonical data that generates the element under the multilinear tensoring operation.

Formal Definition

Let V1, V2, …, Vn be vector spaces over a common field F, and let V1 ⊗ V2 ⊗ ⋯ ⊗ Vn denote their tensor product. An element t of this tensor product is called a pure tensor (or simple tensor, or decomposable tensor) if there exist vectors vi in Vi for each index i such that

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

The Tensor Pure Element Factor List associated with t is the ordered tuple

F (t) = (v_{1}, v_{2}, \dots, v_{n})

The list is indexed by the same ordering as the factor spaces in the tensor product, so the position of each vector in the list identifies unambiguously which vector space it belongs to.

Existence and Non-Uniqueness

Every pure tensor admits at least one factor list by definition, but that list is never the only one that reproduces the element. Multilinearity of the tensor product guarantees that rescaling any factor by a nonzero scalar while inversely rescaling another factor leaves the resulting tensor unchanged.

Scalar Redistribution

For any nonzero scalar λ in F and any two indices i ≠ j, the factor lists

(\dots, v_{i}, \dots, v_{j}, \dots)

and

(\dots, λ v_{i}, \dots, λ^{- 1} v_{j}, \dots)

produce the identical pure tensor. This is the principal source of non-uniqueness: the factor list is determined by the element only up to this scalar redistribution across positions, together with any relation forced by zero factors.

Degenerate Lists

If any single entry of the factor list is the zero vector of its space, the entire pure tensor collapses to the zero element, regardless of the other entries. In that degenerate case, infinitely many distinct factor lists — differing arbitrarily in every non-zero-forced position — all map to the same zero tensor, so the factor list carries essentially no information beyond the fact of degeneracy.

The Factor List as a Function of the Tensor

Because a general element of V1 ⊗ ⋯ ⊗ Vn is a finite sum of pure tensors and need not itself be pure, the factor list is properly understood as a partial construction: it is defined pointwise on the subset of pure elements, not on the whole tensor product space. Determining whether an arbitrary element is pure, and if so recovering a representative factor list, is a nontrivial computational problem in general, closely related to the notion of tensor rank.

Relationship to Tensor Rank

The tensor rank of an element t is the minimal number of pure tensors whose sum equals t. An element has rank at most one exactly when it is pure, in which case its factor list is precisely the data described here. Elements of rank two or greater do not possess a single factor list; instead they are described by a sum of several factor lists, one per term in a minimal decomposition.

Canonical Normalization

To reduce the ambiguity introduced by scalar redistribution, a canonical factor list can be chosen by fixing a normalization convention, such as requiring every factor except the last to have unit norm with respect to a chosen basis or inner product, and absorbing all scale into the final position. Under such a convention, the factor list becomes unique up to simultaneous sign or phase changes across the tuple, depending on the base field.

Coordinates and Component Expansion

When each space Vi is finite-dimensional with a chosen basis of di elements, each factor vi expands in coordinates as

v_{i} = \sum_{k_{i} = 1}^{d_{i}} c_{i, k_{i}} e_{k_{i}}^{(i)}

Substituting these expansions into the tensor product and distributing over multilinearity expresses the pure tensor as a sum over basis tensors, with each coefficient equal to the product of the corresponding component from every entry of the factor list. This shows explicitly how the factor list generates the full coordinate array of the pure tensor from only d1 + d2 + ⋯ + dn numbers, in contrast to the d1 × d2 × ⋯ × dn numbers needed to specify a generic tensor of the same shape.

Computational and Structural Role

The factor list is the standard representation used whenever an algorithm or a proof needs to manipulate a pure tensor without expanding it into full coordinate form. Its compactness underlies several structural uses.

Storage Efficiency

Because a pure tensor is fully determined by its factor list, storing the list requires space linear in the sum of the factor dimensions rather than the product. This is the same principle that motivates low-rank tensor decomposition methods in numerical linear algebra, where a general tensor is approximated by a short sum of pure tensors, each carried by its own factor list.

Multilinear Map Evaluation

If φ : V1 × V2 × ⋯ × Vn → W is a multilinear map and φ̂ : V1 ⊗ ⋯ ⊗ Vn → W is its unique linear extension guaranteed by the universal property of the tensor product, then evaluating φ̂ on a pure tensor reduces to evaluating φ directly on the entries of the factor list:

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

This identity is the operational reason factor lists matter: any linear computation on a pure tensor can be carried out directly on its factors, without ever materializing the full tensor.

Tensor Contraction and Products

When two pure tensors are combined by contraction along matching indices, the operation acts entry-by-entry on the corresponding positions of their factor lists, replacing the contracted pair of factors with their inner product (a scalar) and leaving the remaining factors untouched. This locality is what makes factor-list representations attractive in tensor network computations, where contractions between large arrays are reduced to a sequence of small operations on individual factors.

Illustrative Diagram

The factor list occupies the left side of this diagram: the individually stored vectors from each factor space, each with its own dimension, expand under the tensor product into the single element on the right, whose apparent size is the product of the factor dimensions even though its information content is only their sum.