4.3.4 Tensor Trilinear Slotwise Linearity

Tensor Trilinear Slotwise Linearity describes multilinearity in three tensor slots, ensuring linearity per input while preserving algebraic structure.

Tensor Trilinear Slotwise Linearity is the unifying condition that defines a trilinear map: the requirement that the map be linear in each of its three arguments separately whenever the other two are held fixed, without any requirement of linearity across the arguments jointly. For a map $T : V \times W \times X \to U$ , slotwise linearity consists of three independent conditions, one for each argument slot, and it is precisely these three conditions taken together, rather than any single global linearity property, that constitute trilinearity.

The Three Component Conditions

First Slot Linearity

For fixed $w \in W$ , $x \in X$ :

T (α v_{1} + β v_{2}, w, x) = α T (v_{1}, w, x) + β T (v_{2}, w, x)

Second Slot Linearity

For fixed $v \in V$ , $x \in X$ :

T (v, α w_{1} + β w_{2}, x) = α T (v, w_{1}, x) + β T (v, w_{2}, x)

Third Slot Linearity

For fixed $v \in V$ , $w \in W$ :

T (v, w, α x_{1} + β x_{2}) = α T (v, w, x_{1}) + β T (v, w, x_{2})

All three conditions are required simultaneously; satisfying only two of them characterizes a map that is linear in two arguments but leaves the third unconstrained, which is not trilinear.

Why Slotwise Linearity Is Not Global Linearity

Failure of Linearity on the Product Space

If $V \times W \times X$ is regarded as a single vector space with componentwise addition and scalar multiplication, a trilinear map is almost never linear on it. Scaling every argument by the same scalar $α$ gives

T (α v, α w, α x) = α^{3} T (v, w, x)

which contradicts homogeneity of degree one required for global linearity unless $T$ is identically zero. Slotwise linearity is thus a strictly weaker, and structurally distinct, property from linearity on the product space.

Homogeneity of Degree Three

The cubic scaling behavior exhibited above is a direct symptom of slotwise linearity: applying each of the three single-slot linearity conditions successively to a common scalar $α$ multiplies the output by $α$ three separate times, once per slot, which is the source of the cubic homogeneity characteristic of every trilinear map.

Coordinate Verification

Multi-Index Sum Structure

Slotwise linearity is transparent in coordinates. With structure constants $c_{i j k}^{l}$ , the trilinear map is expressed as

T (v, w, x) = \sum_{i, j, k} v^{i} w^{j} x^{k} c_{i j k}^{l} u_{l}

which is manifestly a polynomial of degree one in each of $v^{i}$ , $w^{j}$ , and $x^{k}$ individually, and degree three overall, giving an immediate coordinate proof of both slotwise linearity and its cubic homogeneity consequence.

Consequence for Tensor Correspondence

Determination by Basis Vectors

Because slotwise linearity, not joint linearity, governs a trilinear map, the map is completely determined by its values on triples of basis vectors, exactly as many values as there are entries in the structure constant array. This finite determination is what allows every trilinear map on finite-dimensional spaces to be represented faithfully by a rank-three tensor, with no information lost or left unconstrained.

Necessity for the Universal Property

Slotwise linearity in all three arguments simultaneously is precisely the hypothesis required for the universal factorization of a trilinear map through the triple tensor product; had the map been linear only in one or two slots, no such unique factoring linear map $\tilde{T} : V \otimes W \otimes X \to U$ would generally exist.