5.6.1 Tensor Pure Element Factorization

Tensor Pure Element Factorization is the question, for a given pure tensor t in V ⊗ W, of recovering vectors v and w with t = v ⊗ w, together with the precise extent to which such a pair (v, w) is uniquely determined by t.

Existence of a Factorization

Guaranteed Only for Elements Already Known to Be Pure

Factorization presupposes that t is already pure, in the sense of having tensor rank at most 1; for a non-pure element, no pair (v, w) satisfies t = v ⊗ w at all, so factorization as discussed here applies only within the cone of pure tensors described under pure tensor structure, not to a general element of V ⊗ W.

Extraction via the Coordinate Matrix

Relative to bases {e_i} of V and {f_j} of W, a pure element t corresponds to a coordinate matrix C = (c_{ij}) of rank at most 1; such a matrix factors as C = ab^T for column vectors a and b, obtained for instance by taking a to be any nonzero column of C and b_j = c_{ij}/a_i for an index i with a_i ≠ 0, giving v = ∑ a_i e_i and w = ∑ b_j f_j with t = v ⊗ w.

Non-Uniqueness of the Factorization

Scalar Rebalancing Between the Two Factors

For any nonzero scalar c, v ⊗ w = (cv) ⊗ (c^{-1} w), so (v, w) and (cv, c^{-1}w) are two different pairs factoring the same pure element t; this rebalancing is always available whenever t is nonzero, since scaling one factor up and the other down by reciprocal amounts leaves the elementary tensor unchanged by the homogeneity identities already established for the scalar action.

This Is the Only Source of Non-Uniqueness for Nonzero t

If v_1 ⊗ w_1 = v_2 ⊗ w_2 for a nonzero pure element and all four vectors are themselves nonzero, then v_2 = c v_1 and w_2 = c^{-1} w_1 for some nonzero scalar c; this follows from expanding both sides in the coordinate matrix picture, where two rank-one matrices ab^T and a'b'^T are equal precisely when a' and b' are related to a and b by this same reciprocal scalar rebalancing. Factorization of a nonzero pure element is therefore unique up to exactly this one-parameter scalar ambiguity, and no other source of non-uniqueness arises.

The Zero Element Is Degenerate

The zero element of V ⊗ W is pure, factoring as v ⊗ 0 or 0 ⊗ w for any v or w whatsoever, since both give the zero tensor by homogeneity; unlike the nonzero case, there is no single scalar family relating all such factorizations, since the zero vector on one side leaves the other factor entirely unconstrained. Factorization of the zero element is therefore treated as a separate, degenerate case rather than a special instance of the scalar-rebalancing pattern that governs the nonzero case.

Consequences for Working with Pure Tensors

Any Two Valid Factorizations Are Interchangeable Up to Rescaling

Because the only ambiguity in factoring a nonzero pure tensor is the reciprocal scalar rebalancing between the two factors, any statement made about t = v ⊗ w that depends only on this product, such as its image under an induced map f ⊗ g or its role as the argument of a bilinear map after factorization, is unaffected by which particular factoring pair (v, w) is chosen — the two constructions f(v) ⊗ g(w) computed from (v, w) and from (cv, c^{-1}w) agree, since f(cv) ⊗ g(c^{-1}w) = c f(v) ⊗ c^{-1} g(w) = f(v) ⊗ g(w).

Higher-Rank Elements Admit No Analogous Factorization

An element of rank 2 or greater has no representation as a single v ⊗ w at all, so the factorization question addressed here is specific to rank at most 1 and does not generalize to describing the minimal-length sum representation of higher-rank elements, which instead falls under the broader treatment of tensor rank.