Multilinear

Multilinear refers to a property of functions that are linear in each of several arguments when the others are held fixed. More formally, a function f: V1 × V2 × ... × Vk → W between vector spaces over a field F is multilinear if, for every i = 1,...,k, f is linear in the i-th argument while the remaining arguments are treated as constants. When W = F, such a function is called a multilinear form or a k-linear form; otherwise it is simply a multilinear map.

The simplest and most familiar example is bilinear maps, which are linear in each of two inputs. The dot product is bilinear: it is linear in each argument separately. The determinant function is multilinear in the columns (or rows) of a matrix, and is alternating as well.

A central relation in multilinear algebra is with tensor products. Multilinear maps correspond to linear maps out of a tensor product: L^k(V1,...,Vk; W) is naturally isomorphic to Hom(V1 ⊗ ... ⊗ Vk, W). This perspective explains why tensor products generalize the notion of combining multiple vector spaces into a single domain for linearity.

Special cases and variants include symmetric multilinear forms, which correspond to symmetric tensors, and alternating multilinear forms, which correspond to exterior tensors. These play key roles in areas such as differential geometry and representation theory.

Applications of multilinear maps span mathematics and applied fields. They underpin tensor decompositions, multilinear regression, and various algorithms in physics, computer graphics, signal processing, and machine learning, where interactions between multiple vector quantities are modeled through multilinearity.