1.2.29 Tensor Factor Definition

In tensor algebra, a tensor factor is a building block used to construct multidimensional arrays through tensor products.

Tensor Factor Definition is the characterization of each individual vector space that participates in a tensor product construction, together with the corresponding role that each such space, or an element drawn from it, plays inside a simple tensor. In an expression of the form $V \otimes W$ , the spaces $V$ and $W$ are called the tensor factors of the product, and in a simple tensor $v \otimes w$ , the individual vectors $v$ and $w$ are likewise referred to as its factors.

Formal Definition

Given a tensor product of vector spaces

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

over a common field $F$ , each vector space $V_{i}$ for $i = 1, \dots, k$ is called a tensor factor of the product space. Correspondingly, an element of the product that can be written in the simple, or decomposable, form

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

with $v_{i} \in V_{i}$ for each $i$ , is said to have tensor factors $v_{1}, v_{2}, \dots, v_{k}$ , one drawn from each of the participating spaces. The order of the factors matters, since the tensor product distinguishes $v \otimes w$ from $w \otimes v$ whenever these lie in different tensor product spaces, $V \otimes W$ and $W \otimes V$ , even though the two spaces are canonically isomorphic.

Distinguishing Space-Level and Vector-Level Factors

The term tensor factor is used at two related levels, and care is needed to distinguish them.

Space-Level Factors

At the level of the ambient product space, the tensor factors are the vector spaces $V_{i}$ themselves, which determine the structure and dimension of the resulting tensor product space. For example, in $V \otimes V^{, the two space-level tensor factors are V and its dual V^{}, and this pairing of factors is what defines the space of type (1, 1) tensors on V . Vector-Level Factors At the level of an individual simple tensor, the tensor factors are the specific vectors v_{i} chosen from each space that combine to produce that element. Two different simple tensors built from the same space-level factors, such as v \otimes w and v^{'} \otimes w^{'}, generally represent different elements of V \otimes W whenever their vector-level factors differ. Non-Uniqueness of Factor Decomposition A key subtlety of tensor factors is that the decomposition of a simple tensor into its factors is not unique, since the bilinearity relations defining the tensor product allow scalars to be redistributed between factors without changing the resulting tensor: (a v) \otimes (\frac{1}{a} w) = v \otimes w for any nonzero scalar a . Consequently, the individual factors v and w of a simple tensor are not themselves canonically determined by the tensor alone; only their combined product v \otimes w, together with the one-dimensional spans they generate, is an invariant of the element. Factors of Non-Simple Tensors Not every element of a tensor product space possesses a single set of tensor factors, since general elements are finite sums of simple tensors rather than simple tensors themselves: T = \sum_{α = 1}^{N} v_{α} \otimes w_{α} Such a tensor is said to have tensor rank N if N is the smallest number of simple terms needed to express it, and it is called entangled or non-decomposable when N > 1, meaning it has no single well-defined set of tensor factors and must instead be described as a sum of several factor pairs. This distinction between decomposable and non-decomposable elements is central to applications of tensor factor structure outside pure algebra, most notably in the description of composite quantum systems, where a decomposable state corresponds to an unentangled configuration and a non-decomposable one corresponds to an entangled configuration. v \otimes w factor 1 factor 2 Role Within Tensor Algebra Identifying the tensor factors of a space or an element is a basic organizational tool throughout tensor algebra. The type (r, s) of a tensor is defined directly in terms of how many of its space-level factors are copies of V and how many are copies of the dual V^{*}, operations such as contraction act on a specified pair of factors, and index notation assigns one index to each tensor factor so that a component T_{i_{1} \dots i_{s}}^{j^{1} \dots j^{r}} can be read off directly against the corresponding factor of the tensor product space. Understanding tensor factors as the atomic building blocks of both the ambient space and its elements is therefore a prerequisite for describing how tensors are constructed, decomposed, and manipulated in coordinates. 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