4.18 Tensor Multilinear Form Structure

Tensor Multilinear Form Structure is the collection of features that organize scalar-valued multilinear maps, multilinear forms, into a coherent algebraic system: their arity, their representation by coordinate arrays relative to bases, their classification into general, symmetric, and alternating types, and their organization into vector spaces that are themselves naturally paired with tensor products.

The Basic Object

Definition

A multilinear form of arity n on vector spaces V₁, ..., Vₙ over a field F is a map

that is linear in each argument separately, with all other arguments held fixed. The structure surrounding this object concerns how such maps combine, how they are represented, and how they relate to one another as n varies and as the spaces Vᵢ vary.

Arity as an Organizing Parameter

The arity n stratifies multilinear forms into linear forms (n = 1), bilinear forms (n = 2), trilinear forms (n = 3), and so on. Increasing arity does not simply add complexity uniformly: the space of forms of arity n grows multiplicatively with the dimensions of the factor spaces, and new phenomena, such as symmetry under permutation of arguments, only become available once n ≥ 2, since permuting a single argument is trivial.

Coordinate Representation

Arrays Indexed by Multiple Indices

When each Vᵢ is finite-dimensional with a chosen basis {e^{(i)}_{k}}, a multilinear form f is completely determined by the array of scalars

For n = 2 this array is a matrix; for general n it is an n-dimensional array, and multilinearity of f guarantees that the value of f on any tuple of vectors is recovered from this array by summing products of coordinates against the array entries, generalizing the bilinear coordinate formula f(v, w) = ∑ᵢⱼ Tᵢⱼ vᵢ wⱼ.

Change of Basis

Under a change of basis on any factor Vᵢ, the array representing f transforms by contraction with the corresponding change-of-basis matrix along the index associated to that factor, leaving the other indices untouched. This transformation rule is the origin of the classical description of a tensor as "an array of numbers that transforms in a prescribed way under change of basis," with the multilinear form being the basis-independent object underlying that array.

The Vector Space of Multilinear Forms

Closure Under Linear Combination

The set of all multilinear forms of a fixed arity n on fixed spaces V₁, ..., Vₙ is itself a vector space, denoted Multilinear(V₁, ..., Vₙ; F), since a scalar multiple or sum of multilinear forms is again multilinear: if f and g are multilinear and α, β ∈ F, then αf + βg satisfies the same linearity condition in each slot.

Dimension in the Finite-Dimensional Case

When each Vᵢ has finite dimension dᵢ, the space of multilinear forms has dimension d₁ × d₂ × ... × dₙ, matching the number of independent entries in the coordinate array described above, since the values on all basis tuples can be prescribed arbitrarily and then extended uniquely by multilinearity.

Identification with the Dual of a Tensor Product

The Correspondence

Every multilinear form corresponds, via the universal property of the tensor product, to a unique linear functional on V₁ ⊗ ... ⊗ Vₙ, and every linear functional on the tensor product arises this way from a unique multilinear form:

This identification is the reason multilinear form structure and tensor product structure are studied together: every structural feature of one side has a counterpart on the other.

Elementary Tensors of Functionals

In the finite-dimensional setting, (V₁ ⊗ ... ⊗ Vₙ)* ≅ V₁* ⊗ ... ⊗ Vₙ*, and under the correspondence above, an elementary tensor φ₁ ⊗ ... ⊗ φₙ of dual vectors corresponds to the "product form" (v₁, ..., vₙ) ↦ φ₁(v₁)⋯φₙ(vₙ). General multilinear forms correspond to finite sums of such product forms, so that the rank of a multilinear form, viewed as a tensor in the dual spaces, is the minimal number of product forms needed to express it as a sum.

Symmetric and Alternating Substructure

Symmetric Forms

When all the Vᵢ coincide with a single space V, a multilinear form f: V × ... × V → F is called symmetric if its value is unchanged under any permutation of its n arguments. Symmetric forms constitute a subspace of the full space of multilinear forms and correspond to linear functionals on the symmetric power Symⁿ(V), a quotient of V ⊗ ... ⊗ V by the relations identifying permuted elementary tensors.

Alternating Forms

A multilinear form is alternating if it vanishes whenever two of its arguments coincide, equivalently if it changes sign under transposition of any two arguments (in characteristic not equal to two). Alternating forms correspond to linear functionals on the exterior power ⋀ⁿV, a further quotient of the tensor power enforcing antisymmetry, and the determinant is the prototypical example of an alternating multilinear form of full arity on a finite-dimensional space.

General Forms Decompose Neither Fully Symmetric Nor Alternating

A general multilinear form need not be symmetric or alternating, and the full space of multilinear forms strictly contains both the symmetric and alternating subspaces, which typically intersect only in the zero form once the arity exceeds one; understanding a general form's structure often proceeds by examining its behavior under the permutation action on its arguments, which decomposes the space of forms according to the representation theory of the symmetric group.

Operations Building New Forms From Old

Tensor Product of Forms

Given a multilinear form f of arity m on spaces U₁, ..., Uₘ and a multilinear form g of arity k on spaces W₁, ..., Wₖ, the assignment (u₁, ..., uₘ, w₁, ..., wₖ) ↦ f(u₁, ..., uₘ) g(w₁, ..., wₖ) defines a multilinear form of arity m + k, matching the tensor product operation on the corresponding dual functionals.

Contraction and Restriction

Fixing one argument of an n-ary multilinear form to a specific vector produces an (n-1)-ary multilinear form on the remaining arguments, an operation dual to the inclusion of a fixed vector as an elementary tensor factor, and iterating this operation reduces any multilinear form step by step down to individual scalar evaluations.