4.18.1 Tensor Multilinear Form Scalar Output

A tensor multilinear form produces a scalar output by applying multilinearity across input vectors in a structured algebraic framework.

Tensor Multilinear Form Scalar Output is the defining feature of a multilinear form: a multilinear map f: V₁ × ... × Vₙ → F whose codomain is the base field F itself rather than some other vector space W. A multilinear form takes several vector arguments, drawn possibly from different spaces, and linearly combines them, one argument at a time, into a single number.

Defining Feature

Codomain Restricted to the Field

A general multilinear map f: V₁ × ... × Vₙ → W allows W to be any vector space. A multilinear form is the special case W = F, so that

f : V_{1} \times ⋮ \times V_{n} \to F

and every output of f is a scalar. This restriction is not merely cosmetic: because the codomain is one-dimensional over itself, scalar-valued multilinear forms enjoy a richer supply of operations, such as multiplication of two forms to produce another, that are unavailable, or require additional structure, when the codomain is a general vector space W.

Multilinearity Restated for Scalar Output

Fixing all arguments except the i-th, the scalar output must depend linearly on that argument:

f (\dots, α v_{i} + β v_{i}', \dots) = α f (\dots, v_{i}, \dots) + β f (\dots, v_{i}', \dots)

with α, β ∈ F. Since the output already lies in F, this equation is a statement purely within the field, without reference to any external vector space structure on the codomain.

Basic Cases by Arity

Linear Forms

When n = 1, a multilinear form is simply a linear functional φ: V → F, an element of the dual space V*. This is the base case from which higher scalar-valued multilinear forms are built.

Bilinear Forms

When n = 2, a multilinear form f: V × W → F is a bilinear form. If V and W are finite-dimensional with bases {eᵢ} and {fⱼ}, the form is completely determined by the scalars f(eᵢ, fⱼ), which can be arranged into a matrix A with Aᵢⱼ = f(eᵢ, fⱼ), giving the coordinate expression

f (v, w) = v^{T} A w

Trilinear and Higher Forms

For n = 3 and beyond, a multilinear form is determined, in the finite-dimensional case, by its values on all tuples of basis vectors, an array of scalars indexed by n indices, f(e_{i₁}, ..., e_{iₙ}), generalizing the matrix of a bilinear form to a higher-dimensional array.

Relation to the Tensor Product

Forms as Elements of the Dual of the Tensor Product

By the universal property of the tensor product, every multilinear form f: V₁ × ... × Vₙ → F corresponds to a unique linear functional f̃ on V₁ ⊗ ... ⊗ Vₙ, that is, an element of the dual space (V₁ ⊗ ... ⊗ Vₙ)*. This gives the identification

Multilinear forms on V_{1}, \dots, V_{n} ≅ {(V_{1} \otimes \dots \otimes V_{n})}^{*}

showing that the scalar-valued case of the tensor universal property produces exactly the dual space of the tensor product, rather than a space of linear maps into some other target.

Elementary Tensors as Product Functionals

When each Vᵢ is finite-dimensional, (V₁ ⊗ ... ⊗ Vₙ)* is itself naturally identified with V₁* ⊗ ... ⊗ Vₙ*, and under this identification an elementary tensor φ₁ ⊗ ... ⊗ φₙ of linear functionals corresponds to the multilinear form

(v_{1}, \dots, v_{n}) \mapsto φ_{1} (v_{1}) ⋮ φ_{n} (v_{n})

so that scalar-valued multilinear forms built from separate linear functionals correspond precisely to elementary tensors of functionals, and general multilinear forms correspond to sums of such products.

Geometric and Illustrative Examples

The Dot Product

The standard dot product on Rⁿ, f(v, w) = ∑ vᵢwᵢ, is a bilinear form with scalar output, and its associated matrix in the standard basis is the identity matrix.

The Determinant

For an n-dimensional space V, the map sending n vectors to the determinant of the matrix formed by their coordinates is an n-fold multilinear form with scalar output; it is additionally alternating, changing sign under transposition of any two arguments.

The Trace Pairing

On the space of n × n matrices, f(A, B) = tr(AB) is a bilinear form with scalar output, used to identify the space of matrices with its own dual space.

Structural Consequences of Scalar Output

Multiplicativity Between Forms

Scalar output allows two multilinear forms, possibly of different arities, to be multiplied pointwise to produce a new multilinear form of combined arity, a construction that underlies the symmetric algebra of polynomial functions and the exterior algebra of alternating forms; this operation has no direct analogue when the codomain is a general vector space W, since there is no canonical multiplication between two arbitrary elements of W.

Pairing with the Tensor Product

Because a scalar-valued multilinear form is the same data as a linear functional on the tensor product, multilinear forms furnish a systematic way to probe and distinguish elements of the tensor product: two tensors are equal if and only if every scalar-valued multilinear form assigns them, via the induced functional, the same value, provided the forms considered are rich enough to separate points, which holds whenever the dual spaces Vᵢ* themselves separate points of Vᵢ.