Editing Multilinear map (section)

{{Short description|Vector-valued function of multiple vectors, linear in each argument}}
{{For|multilinear maps used in cryptography|Cryptographic multilinear map}}
{{More citations needed|date=October 2023}}

In [[linear algebra]], a '''multilinear map''' is a [[function (mathematics)|function]] of several variables that is [[Linear map|linear]] separately in each variable.  More precisely, a multilinear map is a function

:<math>f\colon V_1 \times \cdots \times V_n \to W\text{,}</math>

where <math>V_1,\ldots,V_n</math> (<math>n\in\mathbb Z_{\ge0}</math>) and <math>W</math> are [[vector space]]s (or [[module (mathematics)|module]]s over a [[commutative ring]]), with the following property: for each <math>i</math>, if all of the variables but <math>v_i</math> are held constant, then <math>f(v_1, \ldots, 
 v_i, \ldots, v_n)</math> is a [[linear function]] of <math>v_i</math>.<ref>{{cite book |author-link=Serge Lang |first=Serge |last=Lang |title=Algebra |chapter=XIII. Matrices and Linear Maps §S Determinants |chapter-url=https://books.google.com/books?id=Fge-BwqhqIYC&pg=PA511 |date=2005 |orig-date=2002 |publisher=Springer |edition=3rd |isbn=978-0-387-95385-4 |pages=511– |volume=211 |series=Graduate Texts in Mathematics}}</ref> One way to visualize this is to imagine two [[orthogonal]] vectors; if one of these vectors is scaled by a factor of 2 while the other remains unchanged, the [[cross product]] likewise scales by a factor of two. If both are scaled by a factor of 2, the cross product scales by a factor of <math>2^2</math>.

A multilinear map of one variable is a [[linear map]], and of two variables is a [[bilinear map]].  More generally, for any nonnegative integer <math>k</math>, a multilinear map of ''k'' variables is called a '''''k''-linear map'''.  If the [[codomain]] of a multilinear map is the [[field of scalars]], it is called a [[multilinear form]].  Multilinear maps and multilinear forms are fundamental objects of study in [[multilinear algebra]].

If all variables belong to the same space, one can consider [[symmetric function|symmetric]], [[Bilinear_form#Symmetric,_skew-symmetric_and_alternating_forms|antisymmetric]] and [[alternating map|alternating]] ''k''-linear maps. The latter two coincide if the underlying [[ring (mathematics)|ring]] (or [[field (mathematics)|field]]) has a [[Characteristic (algebra)|characteristic]] different from two, else the former two coincide.