1.2.22 Trilinear Map Definition

Trilinear Map Definition is the characterization of a trilinear map as a function of exactly three vector arguments, drawn from one or more vector spaces defined over a common scalar field, that is linear in each of the three arguments separately whenever the other two are held fixed. It names the specific three-argument case of a multilinear map, standing between the bilinear map and the general multilinear map of arbitrary arity, and provides one of the smallest settings in which the distinctive combinatorial features of higher multilinear maps — such as multiple independent symmetry classes among the arguments — first become apparent.

The Defining Condition

A function taking three vector arguments and producing a result in some vector space is trilinear if, holding any two of the three arguments fixed, the function is linear in the remaining argument, and this must hold for each of the three choices of which argument is allowed to vary. Concretely, this requires that the function respects addition and scalar multiplication independently in its first argument, in its second argument, and in its third argument, with the other two arguments frozen in each case.

The expression above states linearity of a trilinear map in its third argument, with the first two arguments held fixed; analogous conditions in the first and second arguments, each with the remaining two held fixed, complete the full definition of trilinearity.

Trilinear Maps as a Special Case of Multilinear Maps

A trilinear map is precisely a multilinear map taking three arguments, following the same general pattern established for bilinear maps taking two arguments and extended to any finite number of arguments. Every general fact established for multilinear maps of arbitrary arity — that they correspond, via the tensor product's universal property, to linear maps on the tensor product of the spaces involved; that they may be classified as multilinear forms when their output lies in the scalar field; and that they may possess symmetry properties under permutation of their arguments — applies to trilinear maps as the specific instance with exactly three arguments.

Symmetry Properties of Trilinear Forms

A trilinear form, a trilinear map whose output is a scalar, may exhibit several distinct symmetry patterns not possible with only two arguments. It may be fully symmetric, unchanged under any permutation of its three arguments; fully alternating, changing sign under every transposition of two arguments; or symmetric or antisymmetric in only some pair of its arguments while behaving differently with respect to the third. This richer structure of possible symmetries, first appearing at the trilinear level, foreshadows the more elaborate symmetry classifications that become necessary once multilinear maps of even higher arity are considered.

Examples

The scalar triple product of three vectors in three-dimensional space, computed as the dot product of one vector with the cross product of the other two, is a classical example of a trilinear form, and it is fully alternating: permuting its three arguments cyclically leaves its value unchanged, while swapping any two arguments reverses its sign. A trilinear map can also be constructed by composing a bilinear map with a further linear dependence on a third argument, illustrating how higher multilinear maps are frequently built by successively incorporating additional vector arguments into an already multilinear expression.

Role in Tensor Algebra

A trilinear form corresponds, through the tensor product's universal property, to a linear functional on the triple tensor product of the relevant vector spaces, and therefore to a covariant tensor of rank three. Trilinear maps whose arguments are drawn from a mixture of a vector space and its dual correspond to mixed tensors of total rank three, illustrating, at a slightly higher level of complexity than the bilinear case, how the general classification of tensors by type continues to be governed by the same universal property that links multilinear maps of every arity to elements of the appropriate tensor product space.