1.2.34 Decomposable Tensor Definition

A decomposable tensor is a tensor expressible as an outer product of vectors, central to tensor decomposition in algebra.

Decomposable Tensor Definition is the characterization of a tensor within a tensor product space that admits a factorization into a single product of one vector from each contributing factor space. The term decomposable tensor names the same underlying class of objects as simple, elementary, and pure tensors, but it foregrounds the operative question of factorization: whether a given tensor can be broken down, or decomposed, into its constituent vector factors, as opposed to requiring a genuine sum of several such factorizations.

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}

be their tensor product space. A tensor $T \in W$ is decomposable if there exist vectors $v_{i} \in V_{i}$ , for $i = 1, \dots, k$ , satisfying

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

A tensor for which no such factorization exists is called indecomposable, or non-decomposable, and every element of $W$ that is not itself decomposable must be expressed instead as a finite sum of decomposable tensors, since decomposable tensors span the entire tensor product space.

Decomposability as a Closed Condition

The property of being decomposable can be tested by explicit polynomial equations on the coordinates of a tensor once a basis is fixed for each factor space, since the map sending a tuple of vectors to their tensor product is polynomial in the coordinates of the vectors, and the image of this map — the set of decomposable tensors — is cut out by the vanishing of certain polynomials in the coordinates of $T$ . Algebraically, the set of decomposable tensors together with the zero tensor forms the affine cone over the Segre variety associated to the factor spaces, and this variety-theoretic viewpoint is what allows decomposability to be studied using the tools of algebraic geometry rather than only through direct attempts at factorization.

The Two-Factor Case

For a tensor product of two finite-dimensional spaces $V \otimes W$ , decomposability reduces to a familiar linear-algebraic condition. Writing a tensor in coordinates as a matrix $T_{i j}$ , the tensor is decomposable if and only if

rank (T_{i j}) \leq 1

since a matrix of rank at most one is exactly the coordinate expression of an outer product of a single column vector and a single row vector, which is precisely a decomposable tensor. This reduces the general question of decomposability, in the two-factor case, to an ordinary rank computation on a matrix.

Decomposition of Non-Decomposable Tensors

Tensor Rank as a Measure of Non-Decomposability

Since every tensor is a finite sum of decomposable tensors, a natural way to measure how far a given tensor is from being decomposable is to ask for the smallest number of decomposable terms needed to express it. This minimal number is the tensor rank of $T$ ,

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

with $R$ minimal, and a decomposable tensor is precisely one for which $R \leq 1$ . Such an expression as a sum of $R$ decomposable terms is often called a rank decomposition of $T$ .

Difficulty Beyond Two Factors

For tensor products of two factors, computing tensor rank and finding a rank decomposition reduce to standard techniques of linear algebra, such as singular value or eigenvalue-based methods. For tensor products of three or more factors, however, both the value of the tensor rank and the associated decomposition become substantially harder to determine in general, and no closed-form analogue of matrix rank exists; this gap between the well-understood two-factor case and the significantly harder higher-order case is a central theme in the study of decomposable tensors and their generalizations.

Relation to Applications

The distinction between decomposable and indecomposable tensors underlies practical tensor decomposition methods used across data analysis, signal processing, and quantum information. Approximating a high-dimensional data tensor by a sum of a small number of decomposable tensors — a low-rank decomposition — is a standard technique for compressing and interpreting multidimensional data, and in quantum mechanics, decomposability of a state vector in a composite Hilbert space corresponds exactly to the absence of entanglement between the subsystems described by that state.

Role Within Tensor Algebra

Decomposable tensors provide the generative core of any tensor product space: they span the space by finite linear combination, they are the natural elements on which multilinear maps and the universal property of the tensor product are first defined, and the extent to which a tensor departs from decomposability, as quantified by its tensor rank, gives a fundamental invariant used to classify and organize the elements of tensor product spaces throughout tensor algebra.