4 Multilinear Maps

Multilinear Maps is the term for functions of several vector-space arguments that are linear separately in each argument when the remaining arguments are held fixed, generalizing the notion of a linear map to functions of more than one variable and forming the foundational class of objects from which the tensor product, tensors of arbitrary rank, and the theory of covector pairings are all constructed.

Definition and Basic Examples

Formal Definition

A function T from the Cartesian product of vector spaces V_1 times V_2 times cdots times V_k to a vector space W is called multilinear, or k-linear, if for every index i and every fixed choice of the remaining arguments, T is a linear function of its i-th argument.

When k = 1 this reduces to ordinary linearity, and when k = 2 the function is called bilinear, the case most familiar from inner products, matrix multiplication viewed as a pairing, and the pairing bracket between a vector space and its dual.

Elementary Examples

The determinant of an n by n matrix, viewed as a function of its n column vectors, is multilinear, since expanding the determinant along any single column is linear in that column while the others are fixed. The pairing bracket between a covector and a vector is bilinear, being linear in the covector argument and linear in the vector argument separately. The dot product on a real vector space is a bilinear map to the scalars, and matrix multiplication, viewed as a function of a pair of matrices producing a third, is bilinear over the appropriate spaces of matrices.

Multilinear Maps and Covariant Tensors

Covariant Tensors as Multilinear Functionals

A covariant tensor of rank k on a vector space V is defined, in the multilinear-map formulation, as a k-linear map from V times cdots times V, k copies, to the base field F.

This identification recovers the pairing bracket notation for the case k = 1, a single covector, and generalizes it directly: a rank-k covariant tensor is precisely a k-linear functional, evaluated by feeding it a k-tuple of vectors, exactly as a single covector is evaluated by feeding it one vector.

Multilinearity Versus Linearity on the Product Space

It is essential to distinguish multilinearity of T on V_1 times cdots times V_k from ordinary linearity on the direct sum or product of the V_i as a single vector space. A linear map on the direct sum V_1 oplus cdots oplus V_k respects addition across the entire tuple simultaneously, whereas a multilinear map need not be linear when several arguments vary at once; for instance the bilinear map (u, v) maps to u dot v is not linear in the pair (u, v) jointly, since scaling both arguments by c scales the output by c-squared rather than by c. This distinction is precisely what necessitates the tensor product construction rather than a simple direct sum when representing multilinear maps as linear maps.

The Universal Property and the Tensor Product

Linearizing Multilinear Maps

The tensor product V_1 tensor cdots tensor V_k is constructed to solve a universal problem: it is the vector space through which every multilinear map out of V_1 times cdots times V_k factors uniquely as a linear map.

where the tensor product map sends a tuple (v_1, ..., v_k) to the elementary tensor v_1 tensor cdots tensor v_k, and T-tilde is a uniquely determined linear map from the tensor product space to the target space W making the diagram commute. This universal property is the precise sense in which the tensor product converts multilinear algebra into ordinary linear algebra: every fact about multilinear maps on V_1, ..., V_k can be reformulated as a fact about linear maps on the single space V_1 tensor cdots tensor V_k.

Multilinear Maps as the Genesis of Tensor Algebra

Because every tensor of type (p, q) is, by the universal property, equivalent to a specific multilinear map taking p covectors and q vectors as arguments and returning a scalar, the entire apparatus of tensor algebra, contraction, tensor products, symmetrization, can be phrased either in the language of multilinear maps or in the language of elements of iterated tensor product spaces, with the two languages related by the canonical isomorphism supplied by the universal property.

Symmetric and Alternating Multilinear Maps

Symmetric Multilinear Maps

A multilinear map T is called symmetric if its value is unchanged under any permutation of its arguments.

for every permutation sigma of the indices 1 through k. Symmetric bilinear maps generalize quadratic forms and inner products, and symmetric multilinear maps of higher rank underlie the theory of symmetric tensors, quotients of the tensor product space by the relations identifying permuted elementary tensors.

Alternating Multilinear Maps

A multilinear map T is called alternating if it vanishes whenever two of its arguments coincide, a condition equivalent, over fields of characteristic not two, to T changing sign under transposition of any two arguments. The determinant is the prototypical alternating multilinear map, and alternating multilinear maps of rank k correspond to elements of the k-th exterior power of the dual space, the antisymmetric analogue of the tensor product that underlies the theory of differential forms and volume forms.

Composition and Naturality

Precomposition with Linear Maps

If T is a k-linear map on V_1 times cdots times V_k and f_1, ..., f_k are linear maps into V_1, ..., V_k respectively from spaces U_1, ..., U_k, the composite defined by precomposing each argument is again multilinear on U_1 times cdots times U_k. This precomposition operation is the direct generalization, to arbitrary rank, of the pullback of a single covector along a linear map, and it shows that pullback of covariant tensors of any rank along a linear map f is obtained by precomposing the multilinear functional with f in every one of its slots simultaneously, preserving multilinearity at each step of the construction.