4.1.4 Tensor Multilinear Map Output Structure

Tensor Multilinear Map Output Structure describes how tensor products of vectors yield structured outputs through multilinear relationships in algebraic frameworks.

Tensor Multilinear Map Output Structure is the description of the value a tensor's multilinear map produces once every argument slot has been filled, including the algebraic behavior of that output under linearity in each slot, its dependence on the choice of basis, and the way partial filling of slots yields structured outputs of lower valence rather than a single scalar.

The Output for Fully Saturated Arguments

Scalar Output

When every slot of a type (p, q) tensor's multilinear map is filled, with p covectors and q vectors supplied, the output is a single real number,

T (ω^{1}, \dots, ω^{p}, v_{1}, \dots, v_{q}) \in ℝ

obtained by expanding each argument in a basis and summing the product of components against the tensor's own components, as dictated by multilinearity. This scalar output is independent of the basis chosen to perform the expansion, even though the intermediate component values are basis-dependent.

Linearity of the Output in Each Slot

The output responds linearly to changes in any single argument while the others are held fixed. For the k-th vector slot,

T (\dots, a v + b w, \dots) = a \cdot T (\dots, v, \dots) + b \cdot T (\dots, w, \dots)

which is not merely a property of the output value but the defining constraint that makes the assignment of arguments to outputs a multilinear map in the first place.

Structured Output from Partial Saturation

Currying and Lower-Valence Output

Supplying fewer than all p + q arguments leaves an output that is itself a multilinear map on the unfilled slots, rather than a scalar. Fixing q vectors of a type (1, q) tensor, for instance, leaves a linear functional on the remaining covector slot, which is itself an element of V, since (V*)* is naturally identified with V in finite dimensions:

T (\cdot, v_{1}, \dots, v_{q}) \in {(V^{*})}^{*} ≅ V

This behavior is the output-structure justification for treating a tensor as an operator that reduces valence step by step as arguments are supplied, terminating in a scalar only when every slot has been used.

Output as a Vector for Mixed Tensors

More generally, filling all covector slots but leaving r vector slots unfilled in a type (p, q) tensor with r ≤ q leaves an output that is a type (0, r) tensor, an element of the r-fold tensor product of V. Symmetrically, leaving covector slots unfilled while saturating all vector slots leaves a type (s, 0) tensor, an element of the s-fold tensor product of V*. The output type is therefore fully determined by which slots remain open, mirroring the tensor's own valence structure restricted to those slots.

Basis Dependence and Invariance of the Output

Coordinate Expression of the Output

In coordinates, the scalar output for fully saturated arguments is computed by the contraction

T (ω, v) = T_{j}^{i} ω_{i} v^{j}

for a type (1, 1) tensor, where ω_i are the components of ω and v^j are the components of v in dual bases. Although the individual factors T^i_j, ω_i, and v^j all depend on the chosen basis, their fully contracted combination does not, which is the coordinate reflection of the output's basis-independence.

Output Consistency Under Change of Basis

If the basis is changed by an invertible matrix A, the components of T, ω, and v all transform by compensating factors of A and its inverse, and these factors cancel exactly in the contracted sum, leaving the scalar output unchanged. This cancellation is not incidental; it is precisely the condition that a candidate multilinear map must satisfy to define a genuine, coordinate-independent tensor rather than a basis-dependent array of numbers.

Output Under Composition with Linear Maps

Output After Pullback

If T is a type (0, q) tensor on W and f: V → W is linear, the pullback f*T is defined so that its output on vectors in V matches the output of T on their images under f,

(f^{*} T) (v_{1}, \dots, v_{q}) = T (f (v_{1}), \dots, f (v_{q}))

so the output structure of the pulled-back tensor is entirely inherited from the output structure of the original tensor evaluated after transformation of the arguments, guaranteeing that pullback preserves scalar output values exactly along corresponding argument tuples.