multilinear map

Take the idea of a bilinear map and add more input slots. A multilinear map is a machine with k slots that behaves linearly in each one when the others are held fixed. Adjusting any single argument scales or adds in a perfectly proportional way, even though the slots together interact in a richly nonlinear fashion.

Formally, given R-modules M_1, ..., M_k and P, a map f: M_1 × ... × M_k -> P is k-multilinear if, for each index i, fixing all coordinates except the i-th yields an R-linear map M_i -> P. For k = 1 this is just a linear map, for k = 2 a bilinear map; the term k-linear is common. When all M_i equal a fixed module M and P = R, one speaks of a multilinear form of degree k.

Multilinearity is the natural setting for determinants, volume, and tensor calculus: the determinant of an n × n matrix is the unique alternating n-linear form on its columns sending the identity to 1. Just as bilinear maps are linearized by M ⊗ N, every k-multilinear map out of M_1 × ... × M_k factors uniquely through a linear map out of the k-fold tensor product M_1 ⊗ ... ⊗ M_k.