1.2.32 Simple Tensor Definition

A simple tensor is a direct product of vectors, encoding multilinear relationships in algebraic structures.

Simple Tensor Definition is the characterization of a tensor of rank at most one within a tensor product space: an element that can be written as a single product of one vector from each factor space, without requiring any sum of multiple such products. The term simple tensor is used interchangeably with elementary tensor, decomposable tensor, and pure tensor across different areas of mathematics and physics, and it names the most basic and computationally tractable class of elements a tensor product space contains.

Formal Definition

Let $V_{1}, \dots, V_{k}$ be vector spaces over a field $F$ , and let $W = V_{1} \otimes \dots \otimes V_{k}$ denote their tensor product space. An element $T \in W$ is called a simple tensor if there exist vectors $v_{i} \in V_{i}$ for each $i$ such that

T = v_{1} \otimes v_{2} \otimes \dots \otimes v_{k}

The zero vector of $W$ is conventionally treated as a simple tensor of rank zero, since it is produced by the product whenever any single factor $v_{i}$ is itself the zero vector of $V_{i}$ . Every nonzero simple tensor, by contrast, is said to have tensor rank exactly one.

The Set of Simple Tensors as an Algebraic Variety

Unlike most subsets encountered in linear algebra, the set of simple tensors inside $W$ is not a vector subspace, since the sum of two simple tensors is generally not itself simple. It is instead a cone: closed under scalar multiplication, since $a (v_{1} \otimes \dots \otimes v_{k}) = (a v_{1}) \otimes \dots \otimes v_{k}$ , but not closed under addition. Viewed algebraically, the set of simple tensors forms the affine cone over a projective variety known as the Segre variety, cut out by the vanishing of certain polynomial equations in the coordinates of $T$ . This variety structure is what allows the property of being simple to be tested by algebraic conditions on the components of a tensor, rather than only by attempting an explicit factorization.

Component Test for Simplicity in Two Factors

For a tensor product of two finite-dimensional spaces $V \otimes W$ , an element with components $T_{i j}$ relative to chosen bases is simple exactly when all of its $2 \times 2$ minors vanish,

T_{i j} T_{l m} - T_{i m} T_{l j} = 0

for every choice of indices $i, l$ and $j, m$ , which is exactly the condition that the matrix $T_{i j}$ has rank at most one. This gives a finite, checkable criterion for simplicity in the two-factor case, reducing the question of whether a tensor is simple to an ordinary linear-algebraic rank computation.

Non-Simple Tensors and Tensor Rank

A tensor that is not simple is nonetheless always expressible as a finite sum of simple tensors, since the simple tensors span the whole space $W$ . The tensor rank of a general element $T$ is defined as the minimum number $N$ of simple tensors needed in such a sum,

T = \sum_{α = 1}^{N} v_{1}^{α} \otimes \dots \otimes v_{k}^{α}

and a simple tensor is, by this measure, precisely a tensor of rank at most one. This notion of tensor rank generalizes the familiar matrix rank, since in the two-factor case tensor rank and matrix rank coincide exactly, but for three or more factors the two notions diverge in behavior, and determining tensor rank becomes substantially harder than the two-factor case.

Applications Beyond Pure Algebra

The distinction between simple and non-simple tensors carries direct meaning outside abstract algebra. In quantum mechanics, a simple tensor in the tensor product of two Hilbert spaces describing a composite system corresponds to an unentangled, or product, state, in which each subsystem can be assigned a definite state of its own; a non-simple tensor corresponds to an entangled state, in which the subsystems cannot be described independently. In numerical multilinear algebra and data analysis, expressing a data tensor as a sum of few simple tensors, known as a rank decomposition, is a central technique for compressing and interpreting high-dimensional data.

Role Within Tensor Algebra

Simple tensors serve the same generative role for tensor product spaces that basis vectors serve for ordinary vector spaces: they are not the only elements of interest, but every element is built from them, and many structural properties — such as the type of a tensor, or whether a given multilinear map factors through the tensor product — are most naturally defined first on simple tensors before being extended to the whole space by linearity. Understanding which tensors are simple, and by how much a given tensor fails to be simple as measured by its rank, is therefore one of the earliest and most consequential structural questions within the study of any tensor product space.