5.2.2 Tensor Product Element Area

Explore how tensor product elements define areas in multilinear algebra through their bilinear relationships and geometric interpretations.

Tensor Product Element Area is the detailed treatment of individual elements of V ⊗ W, covering the precise sense in which an element is decomposable, an explicit example of a non-decomposable element, the notion of tensor rank, and the relations that govern when two different sums of decomposable elements name the same element.

Decomposable Elements

Definition

An element t of V ⊗ W is decomposable, or pure, if there exist single vectors v in V and w in W with t = v ⊗ w. Every basis element e_i ⊗ f_j of the induced basis is decomposable, but decomposability does not require using basis vectors specifically — any v and w at all give a decomposable v ⊗ w.

A Non-Decomposable Example

Take V = W = F² with bases {e_1, e_2} and {f_1, f_2}, and consider t = e_1 ⊗ f_1 + e_2 ⊗ f_2. Suppose, for contradiction, t = (a_1 e_1 + a_2 e_2) ⊗ (b_1 f_1 + b_2 f_2); expanding by bilinearity gives coefficients a_1 b_1 = 1, a_2 b_2 = 1, a_1 b_2 = 0, a_2 b_1 = 0 on the basis elements e_i ⊗ f_j. From a_1 b_1 = 1, both a_1 and b_1 are nonzero, so a_1 b_2 = 0 forces b_2 = 0; but then a_2 b_2 = 1 is impossible. No choice of a_1, a_2, b_1, b_2 satisfies all four equations, so t is not decomposable.

Tensor Rank

Definition as Minimal Sum Length

The tensor rank of an element t in V ⊗ W is the smallest number r such that t can be written as a sum of r decomposable elements,

t = \sum_{i = 1}^{r} v_{i} \otimes w_{i}

Rank 0 is reserved for the zero element, rank 1 is exactly the decomposable elements, and every element of V ⊗ W has some finite rank at most min(dim V, dim W) because the coordinate array c_{ij} from the induced basis expansion can always be summed one nonzero entry at a time.

Rank Bounded by Matrix Rank

Identifying V ⊗ W with m × n matrices via the coordinate array c_{ij} relative to fixed bases, the tensor rank of an element equals the ordinary matrix rank of its coordinate array; this identification is what makes tensor rank for two-factor tensor products computable by row reduction, in contrast to tensor products of three or more factors, where no such reduction to linear algebra applies.

Relations Between Representations

Non-Uniqueness of Sum Representations

The equality v ⊗ w + v ⊗ w = v ⊗ (2w) shows a rank-1 element written with an apparently longer sum on one side; more generally, any relation derivable from bilinearity — distributing sums, absorbing scalars into either factor — produces a different-looking sum representing the same element.

Equality Is Governed by the Quotient Relations

Two formal sums of decomposable elements represent the same element of V ⊗ W exactly when their difference lies in the relation subspace R used in the construction; in practice, this is checked not by referring to R directly but by expanding both sums in the induced basis {e_i ⊗ f_j} and comparing coefficients, which is finite and mechanical once bases are fixed.