5.7 Tensor Simple Tensor Structure

Tensor Simple Tensor Structure is the body of concepts describing simple (elementary, rank-one, decomposable) tensors within tensor product theory — how they are written, how they are recognized among general tensors, how their factors can be separated back out, and how they relate structurally to the tensors of higher rank that they combine to produce.

Defining Feature of Simple Tensors

A tensor t in a tensor product V1 ⊗ V2 ⊗ ⋯ ⊗ Vn of vector spaces over a field F is simple if it can be written as a single chain of tensored factors,

with vi in Vi for every index i. Simplicity is a property of the element t, not of the notation used to write it: an element either does or does not admit such a factorization, and the entire structure discussed here concerns how that binary property interacts with the algebra and geometry of the ambient tensor product space.

The Three Pillars of Simple Tensor Structure

Study of simple tensors organizes naturally into three closely related questions, each corresponding to a distinct operation one might want to perform on a candidate tensor.

Product Form

The product form is the notational device — the explicit chain v1 ⊗ v2 ⊗ ⋯ ⊗ vn — used to write down a simple tensor once its status as simple, and its factors, are known. Product form is the language in which the universal property of the tensor product is stated, since multilinear maps are defined on tuples of factors and only extend to the full tensor product space by linearity across sums of such forms.

Recognition

Recognition is the decision problem of determining, from a coordinate presentation of an arbitrary tensor, whether that tensor is simple. In the two-factor case this reduces to a rank-at-most-one test on a matrix; in the general n-factor case it requires checking rank-one conditions across every flattening of the tensor and verifying that the resulting candidate factors are mutually consistent.

Factor Separation

Factor separation is the constructive counterpart to recognition: given a tensor already known (or found) to be simple, separation recovers an explicit tuple of factor vectors reproducing it. Separation is only meaningful once recognition has succeeded, and it is inherently non-unique, since scalar weight can be redistributed among the factors without changing their product.

Non-Linearity of the Simple Tensor Set

The collection of all simple tensors inside a tensor product space does not form a linear subspace, even though it is closed under scalar multiplication.

Closure Under Scaling but Not Addition

Scaling a simple tensor by any scalar simply rescales one factor and preserves simplicity. Summing two simple tensors, by contrast, generally produces a tensor of rank two rather than a new simple tensor, so the set of simple tensors is a curved, cone-like subset — an algebraic variety cut out by the vanishing of the rank-one minors of the tensor's flattenings — rather than a flat subspace.

Consequence for Decomposition

Because simple tensors do not close under addition, an arbitrary tensor's expression as a sum of simple tensors (its rank decomposition) is not unique in general, and the minimal number of terms required — the tensor rank — becomes the natural invariant measuring how far a tensor sits from being simple itself.

Simple Tensors as Generators

Despite their nonlinear structure as a set, simple tensors generate the entire tensor product space linearly: every element of V1 ⊗ ⋯ ⊗ Vn is a finite sum of simple tensors, and this generating role underlies the standard construction of the tensor product as a quotient of a free vector space on symbols of exactly this simple form.

Basis Tensors as a Distinguished Family

When each factor space is finite-dimensional, choosing bases produces a finite family of simple tensors — one for each combination of basis vectors across the factors — that forms an actual basis of the tensor product space, giving the most economical possible generating family drawn entirely from simple tensors.

Role in Applications

Simple tensor structure underlies practical tensor decomposition methods used throughout numerical linear algebra, signal processing, and machine learning.

Rank Decomposition Algorithms

Algorithms such as canonical polyadic decomposition express a data tensor as a minimal sum of simple tensors, treating each term as an independent factor separation problem coupled across terms by the overall approximation objective. The well-posedness and stability of such algorithms depend directly on properties of the nonlinear simple-tensor set, including the fact that the set of tensors of bounded rank can fail to be closed.

Tensor Networks

In tensor network formalisms, individual node tensors are frequently taken to be simple, or built as short sums of simple tensors, so that large computations decompose into local operations on individual factors — a direct computational payoff of simple tensor structure's compact, factorized representation.

Illustrative Diagram

The three overlapping regions correspond to the three pillars of simple tensor structure — writing, deciding, and separating — each of which depends on the others to give a complete account of a simple element.