1.8.4 Multilinear Map Tensor Abstraction

Multilinear Map Tensor Abstraction unifies multilinear transformations into tensor structures, providing a framework for generalizing linear algebra to higher dimensions.

Multilinear Map Tensor Abstraction is the process of replacing a function of several vector arguments, linear separately in each one, with a single linear map defined on one combined space, so that the multilinear structure spread across several arguments is absorbed entirely into the structure of the combined space, and questions about multilinear maps of any number of variables reduce uniformly to questions about ordinary linear maps of one variable. It is the abstraction step that explains why tensor products, rather than some other combining device, are the natural home for multilinear phenomena of every arity.

From Several Arguments to One

The Multilinear Starting Point

A k-multilinear map takes k vector arguments, drawn from spaces V_1, ..., V_k, and produces a value in some space X, subject to linearity in each argument separately while the other arguments are held fixed.

β : V_{1} \times \dots \times V_{k} \to X

The Linearized Replacement

Multilinear map tensor abstraction replaces β with a single linear map β̃ defined on the tensor product V_1 ⊗ ⋯ ⊗ V_k, related to β by evaluation on simple tensors, so that all of the multilinear behavior of β is encoded in the ordinary linearity of β̃.

\tilde{β} (v_{1} \otimes \dots \otimes v_{k}) = β (v_{1}, \dots, v_{k})

The General Universal Property

Statement for Arbitrary Arity

The universal property of the tensor product extends directly from the bilinear case, k = 2, to arbitrary k: every k-multilinear map out of V_1 × ⋯ × V_k factors uniquely through the iterated tensor product V_1 ⊗ ⋯ ⊗ V_k by way of a linear map.

Mult (V_{1}, \dots, V_{k}; X) ≅ Hom (V_{1} \otimes \dots \otimes V_{k}, X)

Associativity of the Iterated Product

Because the tensor product is associative up to canonical isomorphism, the iterated tensor product V_1 ⊗ ⋯ ⊗ V_k is unambiguous regardless of how the factors are grouped, which is what allows the multilinear map abstraction to be stated for arbitrary arity without needing to fix a particular bracketing of the product.

Currying as an Alternative Route to the Same Abstraction

Reducing Arity One Step at a Time

An alternative way to linearize a bilinear map is currying: rather than passing directly to a two-fold tensor product, a bilinear map β : V × W → X is reinterpreted as a linear map from V into the space of linear maps from W to X.

Bilin (V, W; X) ≅ Hom (V, Hom (W, X))

Consistency of the Two Routes

Currying and the tensor product route both linearize the same bilinear map, and the isomorphism Hom(V ⊗ W, X) ≅ Hom(V, Hom(W, X)) shows the two abstractions agree, differing only in whether the multilinear information is absorbed into a combined domain space or distributed across a nested sequence of function spaces.

Symmetric and Alternating Multilinear Maps

Restricting to Maps Invariant Under Permutation

When the arguments of a multilinear map all come from the same space V and the map is required to be symmetric under permutation of its arguments, the corresponding linearized map factors further, through the symmetric power Sym^k(V) rather than the full tensor power, since the abstraction respects the extra symmetry constraint imposed on β.

SymMult (V; X) ≅ Hom ({Sym}^{k} (V), X)

Restricting to Alternating Maps

Likewise, an alternating multilinear map, one that changes sign under a transposition of two arguments, factors through the exterior power ）^k(V), giving the same abstraction procedure a third specialization used to construct determinant-like and volume-like quantities.

Why This Abstraction Is the Right One

Uniqueness from the Universal Property

The universal property guarantees that the tensor product is, up to unique isomorphism, the only space through which every multilinear map factors linearly, so multilinear map tensor abstraction is not merely one convenient linearization device among several but the canonical one, in the precise sense that any other space with the same factorization property must be isomorphic to it.

Reduction of Multilinear Algebra to Linear Algebra

Because every multilinear question can be rephrased, through this abstraction, as a question about a single linear map on a tensor product space, the entire theory of multilinear forms, bilinear forms, trilinear forms, and beyond, is absorbed into ordinary linear algebra applied to iterated tensor product spaces.

Diagrammatic Summary

The diagram shows how a multilinear map β of k arguments is linearized into a single linear map β̃ on the tensor product space, the two paths to X agreeing exactly as required by the universal property.