5.6 Tensor Pure Tensor Structure

Tensor Pure Tensor Structure is the study of the set of pure tensors — the decomposable elements v ⊗ w of V ⊗ W, identical to the elementary tensors already introduced but examined here as a set in its own right — covering how that set sits inside V ⊗ W, what algebraic closure properties it does and does not have, and how it is characterized without reference to any particular factorization of its elements.

Terminology: Pure Tensor and Elementary Tensor Name the Same Objects

One Concept, Two Established Names

"Pure tensor" and "elementary tensor" both refer to elements of V ⊗ W expressible as v ⊗ w for single vectors v and w; the term "pure" emphasizes that such an element involves no summation, echoing the language of pure states in contrast to mixtures, while "elementary" emphasizes the role of such elements as the basic building blocks from which every element of V ⊗ W is assembled. Both terms are used throughout tensor algebra depending on context, without a difference in meaning.

The Distinction of Focus From Elementary Tensor Structure

Where the elementary tensor structure topic examines a single elementary tensor's symbolic form, its factors, their order, and the product rule producing it, pure tensor structure instead examines the collection of all such elements as a subset of V ⊗ W, asking what shape that subset has within the ambient vector space.

The Set of Pure Tensors as a Subset of V ⊗ W

Not a Subspace

The set of pure tensors is closed under scalar multiplication — c(v ⊗ w) = (cv) ⊗ w is again pure — but is not closed under addition in general, since the sum e_1 ⊗ f_1 + e_2 ⊗ f_2 of two pure tensors was already shown, under the treatment of tensor rank, to be non-decomposable. This failure of closure under addition is what prevents the pure tensors from forming a subspace of V ⊗ W, despite containing the zero element and being closed under scaling.

A Cone, Not a Linear Subspace

Because it is closed under scalar multiplication but not addition, the set of pure tensors forms a cone: for any pure tensor t and scalar c, ct is pure, but the set does not extend to a subspace by taking sums. This cone structure is the precise algebraic shape occupied by the pure tensors inside V ⊗ W, strictly smaller than any subspace containing it other than V ⊗ W itself once dim V, dim W > 1.

Characterizing Pure Tensors Without Reference to a Specific Factorization

The Rank-One Characterization

An element t of V ⊗ W is pure precisely when its tensor rank is 0 or 1; this characterization is intrinsic to t and does not require exhibiting the specific v and w involved, making it usable as a test for purity even before any factorization has been found.

The Coordinate-Array Characterization in the Finite-Dimensional Case

Relative to bases of V and W, identifying V ⊗ W with m × n matrices via the induced coordinate array, an element is pure exactly when its coordinate matrix has rank at most 1; this reduces the question of purity to ordinary linear-algebraic rank computation once coordinates are fixed, giving a concrete, checkable criterion equivalent to the abstract rank-one characterization above.

Role Within Tensor Product Theory

Marking the Boundary Between Simple and General Elements

Pure tensor structure formalizes the line between the simplest possible elements of V ⊗ W and the general elements built by summing them, a distinction used throughout tensor algebra whenever a result is first established for pure tensors and then extended by linearity to the whole space, exactly as the expansion role of elementary tensors is used to do.