1.2.24 Multilinear Form Definition

A multilinear form is a function that is linear in each of its arguments, generalizing linear forms to multiple variables in tensor algebra.

Multilinear Form Definition is the characterization of a function that takes several vector arguments from one or more vector spaces and returns a scalar, while being linear in each argument separately when all other arguments are held fixed. Such a function is also called a multilinear map or, when the scalar output lies in the base field of the vector spaces involved, a multilinear functional. Multilinear forms generalize the familiar notions of linear functionals, bilinear forms, and determinants into a single unified framework and serve as one of the foundational building blocks of tensor algebra.

Formal Definition

Let $V_{1}, V_{2}, \dots, V_{k}$ be vector spaces over a common field $F$ . A function

T : V_{1} \times V_{2} \times \dots \times V_{k} \to F

is called a multilinear form of degree $k$ , or a $k$ -linear form, if for every index $i$ from $1$ to $k$ , and for every fixed choice of vectors in all the other coordinates, the resulting function of the remaining single variable is linear. That is, for vectors $u, v \in V_{i}$ and scalars $a, b \in F$ ,

T (v_{1}, \dots, a u + b v, \dots, v_{k}) = a T (v_{1}, \dots, u, \dots, v_{k}) + b T (v_{1}, \dots, v, \dots, v_{k})

where the marked coordinate occupies position $i$ and every other coordinate is held fixed. This condition must hold independently for each of the $k$ arguments; linearity in one slot alone does not imply linearity in the others, which is why the definition requires the property to be checked separately, slot by slot.

Special Cases by Degree

Linear Forms

When $k = 1$ , a multilinear form reduces to an ordinary linear functional on a single vector space: a map that is additive and homogeneous in its one argument. Linear forms are the simplest instance of the multilinear concept and generate the dual space of the vector space on which they act.

Bilinear Forms

When $k = 2$ , the definition specializes to a bilinear form: a function of two vector arguments that is linear in each argument when the other is fixed. Familiar examples include inner products, symplectic forms, and the pairing between a vector space and its dual. Bilinear forms may further be classified as symmetric, antisymmetric, or neither, depending on their behavior under exchange of arguments.

Higher-Degree Forms

For $k \geq 3$ , multilinear forms capture structures with no direct two-argument analogue, such as the determinant of a square matrix viewed as an alternating multilinear form of its column vectors, or torsion and curvature-type expressions in differential geometry that require several vector inputs simultaneously.

Symmetry Properties

A multilinear form may possess additional structure beyond multilinearity, determined by how it behaves when its arguments are permuted.

Symmetric Forms

A multilinear form $T$ is symmetric if its value is unchanged under any permutation $σ$ of its arguments:

T (v_{σ (1)}, \dots, v_{σ (k)}) = T (v_{1}, \dots, v_{k})

Alternating (Antisymmetric) Forms

A multilinear form is alternating if it vanishes whenever two of its arguments coincide, which implies that swapping any two arguments negates its value. Alternating multilinear forms are the algebraic foundation of determinants and of exterior algebra, where they appear as the natural objects assigned to volumes, areas, and oriented geometric quantities spanned by several vectors.

General Forms

A multilinear form need not be symmetric or alternating; it may exhibit no permutation symmetry at all, in which case its behavior under argument exchange must be analyzed case by case rather than inferred from a general rule.

Relationship to Tensors

Multilinear forms are intimately connected to the concept of a tensor. A covariant tensor of order $k$ on a vector space $V$ is, by definition, a multilinear form

T : V \times \dots \times V \to F

taking $k$ copies of the same vector space as arguments. This identification means that multilinear forms are not merely analogous to tensors but constitute one of the standard rigorous definitions of what a tensor is: an element of the space of multilinear maps from a product of vector spaces (and their duals, for mixed tensors) to the base field. The set of all $k$ -linear forms on $V$ itself forms a vector space under pointwise addition and scalar multiplication, denoted the space of covariant $k$ -tensors on $V$ .

Representation in Finite Dimensions

When each $V_{i}$ is finite-dimensional with a chosen basis, a multilinear form is completely determined by its values on all combinations of basis vectors. If ${e_{1}, \dots, e_{n}}$ is a basis for $V$ , then a $k$ -linear form on $V$ is fully specified by the $n^{k}$ scalars

T_{i_{1} \dots i_{k}} = T (e_{i_{1}}, \dots, e_{i_{k}})

obtained by evaluating the form on every ordered tuple of basis vectors. The value of the form on arbitrary vectors is then recovered by expanding each argument in the chosen basis and applying multilinearity to distribute the form over the resulting sum, producing an expression in the components of the vectors and the scalars $T_{i_{1} \dots i_{k}}$ . This component representation underlies the index notation used throughout tensor algebra and tensor calculus.

Role in Tensor Algebra

Multilinear forms provide the conceptual bridge between abstract tensor definitions and their concrete manipulation. Operations central to tensor algebra — such as the tensor product, contraction, and change of basis — are most cleanly understood in terms of how multilinear forms act on tuples of vectors and how their component representations transform under a change of basis. The requirement that physical and geometric quantities transform consistently under coordinate changes is, at its core, a statement about the multilinearity of the underlying form, making the definition of multilinear forms a prerequisite for the systematic study of tensors and their applications in geometry, physics, and algebra.