4.22 Tensor Multilinear Map Representation

Tensor Multilinear Map Representation is the collection of equivalent ways a multilinear map can be described concretely, ranging from a basis-free linear map on a tensor product to a basis-dependent array of numbers, each representation suited to different kinds of reasoning about the same underlying multilinear object.

The Basis-Free Representation

As a Linear Map on the Tensor Product

By the universal property of the tensor product, every multilinear map f: V₁ × ... × Vₙ → W corresponds to a unique linear map f̃: V₁ ⊗ ... ⊗ Vₙ → W. This representation requires no choice of basis and is the representation used whenever a fact about f is to be established independent of coordinates, such as an isomorphism or a universal-property argument.

As an Element of a Hom-Space

Equivalently, f can be represented as an element of Hom(V₁ ⊗ ... ⊗ Vₙ, W), the vector space of linear maps between the tensor product and W; this framing is used when the collection of all multilinear maps of a given arity and target is itself studied as a vector space, since Hom(V₁ ⊗ ... ⊗ Vₙ, W) inherits vector space structure directly from W.

The Component Array Representation

Coordinates Relative to Chosen Bases

Once bases for V₁, ..., Vₙ and W are chosen, f is represented by a component array T^{k}_{i₁...iₙ}, giving the coefficient of the k-th basis vector of W in the expansion of f evaluated on basis tuples; for scalar-valued forms, the index k is absent, and T_{i₁...iₙ} alone suffices.

The Matrix Case as a Special Instance

For arity n = 2 with scalar output, the component array reduces to an ordinary matrix, the most familiar and computationally convenient instance of this representation, from which values of f on arbitrary vectors are recovered by matrix multiplication.

The Dual Tensor Representation for Scalar Output

Elements of a Dual Tensor Product

When W = F, the correspondence given by the universal property identifies f with an element of (V₁ ⊗ ... ⊗ Vₙ)*, and in finite dimensions further with an element of V₁* ⊗ ... ⊗ Vₙ*. This representation situates scalar-valued multilinear maps directly within tensor algebra, treating them as tensors of dual vectors rather than as maps.

Rank as a Representation-Independent Invariant

The minimal number of elementary tensors of dual vectors needed to express f under this representation, its tensor rank, matches the matrix rank of f when n = 2, and generalizes this notion to higher arity; rank is an invariant of f computable from any of its representations, though the dual tensor representation makes its definition most direct.

The Curried Representation

Multilinear Maps as Iterated Linear Maps

Fixing one argument slot at a time, f can be represented as a linear map into a space of multilinear maps of one lower arity:

This representation is used when a multilinear map is to be analyzed one slot at a time, or when relating multilinear maps of different arities via partial application.

Choosing Among Representations

Basis-Free for Structural Arguments

Proofs that a construction is well defined, that two constructions agree canonically, or that a map respects some universal property are carried out most cleanly using the basis-free linear-map representation, since basis-dependent bookkeeping would obscure rather than clarify the structural content of such arguments.

Component Arrays for Explicit Computation

Direct numerical evaluation of a multilinear map on specific vectors, or comparison against known formulas such as the determinant or standard inner products, is carried out using the component array representation, since this is the representation compatible with numerical linear algebra software and explicit hand computation.

Dual Tensor Elements for Rank and Decomposition Questions

Questions about how efficiently a scalar-valued multilinear form can be decomposed into simpler pieces, its rank, its expression as a sum of product forms, are naturally phrased using the dual tensor representation, since decomposition into elementary tensors is exactly what tensor rank measures.

Converting Between Representations

From Basis-Free to Component Array

Given the linear map f̃ on the tensor product, choosing bases for the factor spaces and evaluating f̃ on elementary tensors of basis vectors produces the component array directly; this conversion always exists once bases are fixed, since elementary tensors of basis vectors span the tensor product.

From Component Array Back to Basis-Free

Conversely, given a component array and the same choice of bases, the reconstruction formula f(v₁,...,vₙ) = ∑ v₁^{i₁}⋯vₙ^{iₙ} T_{i₁...iₙ} recovers the value of f on any tuple, and hence recovers the basis-free map f itself, since a multilinear map is determined completely by its values on all tuples of vectors.

Consistency Across Conversions

Passing from one representation to another and back again recovers the original object exactly, since every representation is, by the universal property and the spanning property of bases, a complete and faithful description of the same underlying multilinear map; no information is gained or lost by choosing one representation over another, only convenience for the task at hand.

Representation and the Symmetric or Alternating Case

Reduced Component Arrays

When f is additionally symmetric or alternating, its component array representation can be compressed, to non-decreasing index tuples in the symmetric case or to strictly increasing index tuples in the alternating case, reducing the number of independent scalars needed relative to a general multilinear map of the same arity and dimension, a saving directly reflected in the smaller dimension of Symⁿ(V) or ⋀ⁿV compared to the full tensor power V^{⊗n}.