5.6.4 Tensor Pure Element Nonlinear Subset

A nonlinear subset in tensor algebra where pure elements interact through non-linear operations, foundational in advanced mathematical structures.

Tensor Pure Element Nonlinear Subset is the description of the set of all pure (simple, decomposable) tensors inside a tensor product space as a subset that is closed under scalar multiplication but not under addition, and which therefore fails to be a linear subspace despite living inside a vector space. The pure tensors form a cone-shaped, algebraically curved region rather than a flat one, which is why the qualifier "nonlinear" is attached to the subset.

Formal Description

Let T denote the tensor product V1 ⊗ V2 ⊗ ⋯ ⊗ Vn of vector spaces over a field F, and let P be the subset of pure tensors,

P = {v_{1} \otimes \dots \otimes v_{n} ∣ v_{i} \in V_{i}} \subseteq T

P is closed under scalar multiplication: multiplying a pure tensor by any scalar simply rescales one of its factors, so λ·t remains in P whenever t is. P is not, in general, closed under addition: the sum of two pure tensors is typically not itself expressible as a single product of factors, so P is not a vector subspace of T, even though it sits inside one.

Why Sums of Pure Tensors Escape the Subset

The failure of closure under addition is the defining feature that makes P a nonlinear subset, and it can be seen directly from a small example.

A Concrete Failure of Closure

Taking V1 = V2 = F² with basis vectors e1, e2, the tensors e1 ⊗ e1 and e2 ⊗ e2 are both pure. Their sum

e_{1} \otimes e_{1} + e_{2} \otimes e_{2}

cannot be written as a single tensor v ⊗ w for any vectors v and w. When identified with a 2-by-2 matrix, this sum corresponds to the identity matrix, whose determinant is nonzero, whereas every pure tensor of this two-factor form corresponds to a rank-one matrix, whose determinant is always zero. This mismatch is a direct proof that the sum lies outside P.

General Obstruction via Rank

More generally, in the two-factor case, pure tensors correspond exactly to matrices of rank at most one, and the set of rank-at-most-one matrices is known to be closed under scalar multiplication but not under addition, since adding two rank-one matrices can raise the rank to two. This matrix picture is the clearest low-dimensional illustration of why P is nonlinear.

Algebraic Variety Structure

Even though P is not a linear subspace, it is far from an arbitrary subset: it is a well-behaved algebraic variety, describable as the common zero set of a specific family of polynomial equations.

The Segre Variety

In the language of projective and algebraic geometry, the projectivization of P — the set of pure tensors considered up to overall nonzero scalar — is known as the Segre variety, embedded inside the projective space associated with T via the Segre embedding. Membership in P is characterized by the vanishing of all the 2-by-2 minors of appropriate "flattenings" (matricizations) of the tensor, which are polynomial equations of degree two in the coordinates of T.

Dimension of the Nonlinear Subset

If each factor space Vi has dimension di, the space of pure tensors has dimension equal to

(\sum_{i = 1}^{n} (d_{i} - 1)) + 1

which is generically far smaller than the dimension of the ambient tensor product space, the product d1 · d2 · ⋯ · dn. This dimension gap is the geometric reflection of the algebraic fact that P is a thin, curved subset rather than the whole space or a linear slice of it.

Consequences of Nonlinearity

The nonlinear character of the pure tensor subset has direct consequences for how tensors are analyzed and approximated.

Rank as a Measure of Distance from the Subset

Because most tensors are not pure, the tensor rank of an element measures how far, in a combinatorial sense, that element sits from the nonlinear subset P: rank one means membership in P itself, and higher rank means the element requires a sum of several elements of P to represent it. Since P is not closed under addition, this notion of rank is nontrivial and does not collapse to a linear-algebra invariant the way the rank of a single matrix summand would.

Optimization Over a Curved Set

Algorithms that seek the best pure-tensor (rank-one) approximation to a given tensor are optimizing over the nonlinear subset P rather than over a linear subspace. This is qualitatively different from linear least-squares problems: because P is curved and not convex, such optimization problems can have multiple local optima, and the best rank-one approximation is not obtained by any simple linear projection, unlike the analogous problem for ordinary vector subspaces.

Instability Near the Boundary of Rank

The set of tensors of rank at most two or more can fail to be closed even though rank itself is defined via finite sums, a phenomenon tied to the fact that these higher-rank sets are built from unions and closures of copies of the nonlinear subset P. Sequences of tensors of a fixed small rank can converge to a limit of strictly higher rank, an effect with no counterpart in ordinary linear subspace theory and one that originates precisely in the nonlinearity of P.

Illustrative Diagram

The curved line traces the nonlinear subset P inside the flat rectangular region representing the full tensor product space; two marked points on the curve are pure tensors whose straight-line sum, lying off the curve, illustrates the failure of closure under addition.