Handbook of Linear Algebra, 2nd Edition

Chapter 14

Multilinear Algebra

J. A. Dias da Silva

Universidade de Lisboa

Armando Machado

Universidade de Lisboa

14.1 Multilinear Maps

Unless otherwise stated, within this section V, U, and W as well as these letters with sub-scripts, superscripts, or accents, are finite dimensional vector spaces over a field F of characteristic zero.

Definitions:

A map φ from V1 ×···× Vm into U is a multilinear map (m-linear map) if it is linear on each coordinate, i.e., for every v′i, vi ∈, Vi,i = 1,... ,m and for every a ∈ F the following conditions hold:

1. (a) φ(v1,...,vi + ${\text{v}}_i^{\prime }$,...,vm)= φ(v1,...,vi,...,vm)= φ(v1,...,${\text{v}}_i^{\prime }$,...,vm);
2. (b) φ(v1,..., avi,..., vm) = aφ(v1,..., vi,..., vm).

The 2-linear maps and 3-linear maps are also called bilinear and trilinear