1.2.31 Elementary Tensor Definition

Elementary tensors are foundational in tensor algebra, formed by the tensor product of vectors, and form the basis for complex tensor operations.

Elementary Tensor Definition is the characterization of a tensor that can be written as a single tensor product of vectors drawn from each of the participating vector spaces, rather than as a sum of several such products. An elementary tensor, also called a simple tensor, decomposable tensor, or pure tensor, is the most basic kind of element that a tensor product space contains, and every other element of the space is built from finite sums of elementary tensors.

Formal Definition

Let $V_{1}, \dots, V_{k}$ be vector spaces over a field $F$ , and consider the tensor product space

W = V_{1} \otimes V_{2} \otimes \dots \otimes V_{k}

An element $T \in W$ is called an elementary tensor if it can be expressed in the form

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

for some choice of vectors $v_{i} \in V_{i}$ , one from each factor space. This single-term form is what distinguishes an elementary tensor from a general element of $W$ , which is only guaranteed to be expressible as a finite sum of terms of this kind, not as one term alone.

Elementary Tensors Generate the Tensor Product Space

Although not every element of $W$ is elementary, the set of all elementary tensors spans $W$ as a vector space, so that every element $T$ can be written as

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

for some finite $N$ and appropriate choices of vectors. This is a direct consequence of the explicit construction of the tensor product as a quotient of a free vector space on pairs of basis vectors: since a basis of $W$ can always be chosen consisting entirely of elementary tensors formed from basis vectors of each factor space, every element of $W$ reduces to a finite linear combination of elementary tensors.

Non-Elementary Tensors

An element of $W$ that cannot be written as a single elementary tensor, no matter how the sum is rearranged, is called a non-elementary or entangled tensor. The standard example occurs in a two-factor tensor product $V \otimes W$ with basis vectors $e_{1}, e_{2}$ of $V$ and $f_{1}, f_{2}$ of $W$ : the element

T = e_{1} \otimes f_{1} + e_{2} \otimes f_{2}

cannot be written as a single product $v \otimes w$ for any $v \in V$ and $w \in W$ , which can be verified by attempting to solve for such $v$ and $w$ in coordinates and finding no consistent solution. Such an element is said to have tensor rank two, meaning two is the minimum number of elementary tensors needed to sum to it, in contrast to an elementary tensor, which by definition has tensor rank at most one.

Criterion for Elementarity

The Matrix Rank Test in the Two-Factor Case

For a two-factor tensor product $V \otimes W$ of finite-dimensional spaces, an element $T$ expressed in coordinates as a matrix $T_{i j}$ relative to chosen bases is elementary if and only if this matrix has rank at most one. A rank-one matrix can always be written as an outer product of a column vector and a row vector, which is precisely the coordinate expression of an elementary tensor $v \otimes w$ . This connects the abstract notion of elementarity directly to the ordinary linear-algebraic notion of matrix rank.

General Multi-Factor Case

For tensor products of more than two factors, the tensor rank of a general element is defined analogously as the minimum number of elementary tensors needed to sum to it, but unlike the two-factor case, no simple linear-algebraic test analogous to matrix rank determines this number in general, and computing tensor rank for three or more factors is known to be computationally difficult in general.

Role Within Tensor Algebra

Elementary tensors function as the basic building blocks from which every construction in tensor algebra is assembled: bases of tensor product spaces are chosen among them, the universal property of the tensor product is verified on them before being extended by linearity to all tensors, and the type of a general tensor — symmetric, alternating, or of mixed type — is typically defined first for elementary tensors and then propagated to sums by linearity. Recognizing which tensors are elementary and which require a genuine sum of several terms is accordingly one of the first structural questions to address when working with any specific tensor product space.