Multilinear map

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a multilinear map is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> of several variables that is <a href="/facts/Linear_map/Q1PBKNVV">linear</a> separately in each variable. More precisely, a multilinear map is a function

f
        :
        
          V
          
            1
          
        
        ×
        ⋯
        ×
        
          V
          
            n
          
        
        →
        W
        
          ,
        
      
    
    {\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}

where 
 
 
 
 
 V
 
 1
 
 
 ,
 …
 ,
 
 V
 
 n
 
 
 
 
 {\displaystyle V_{1},\ldots ,V_{n}}
 
 (
 
 
 
 n
 ∈
 
 
 Z
 
 
 ≥
 0
 
 
 
 
 {\displaystyle n\in \mathbb {Z} _{\geq 0}}
 
) and 
 
 
 
 W
 
 
 {\displaystyle W}
 
 are <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> (or <a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a> over a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a>), with the following property: for each 
 
 
 
 i
 
 
 {\displaystyle i}
 
, if all of the variables but 
 
 
 
 
 v
 
 i
 
 
 
 
 {\displaystyle v_{i}}
 
 are held constant, then 
 
 
 
 f
 (
 
 v
 
 1
 
 
 ,
 …
 ,
 
 v
 
 i
 
 
 ,
 …
 ,
 
 v
 
 n
 
 
 )
 
 
 {\displaystyle f(v_{1},\ldots ,v_{i},\ldots ,v_{n})}
 
 is a <a href="/facts/Linear_function/Ejvy8CKy">linear function</a> of 
 
 
 
 
 v
 
 i
 
 
 
 
 {\displaystyle v_{i}}
 
. One way to visualize this is to imagine two <a href="/facts/Orthogonal/JVIjYPog">orthogonal</a> vectors; if one of these vectors is scaled by a factor of 2 while the other remains unchanged, the <a href="/facts/Cross_product/uRBJzkcW">cross product</a> likewise scales by a factor of two. If both are scaled by a factor of 2, the cross product scales by a factor of 
 
 
 
 
 2
 
 2
 
 
 
 
 {\displaystyle 2^{2}}
 
.
A multilinear map of one variable is a <a href="/facts/Linear_map/Q1PBKNVV">linear map</a>, and of two variables is a <a href="/facts/Bilinear_map/LH1lHVeP">bilinear map</a>. More generally, for any nonnegative integer 
 
 
 
 k
 
 
 {\displaystyle k}
 
, a multilinear map of k variables is called a k-linear map. If the <a href="/facts/Codomain/MGc43nwQ">codomain</a> of a multilinear map is the <a href="/facts/Field_of_scalars/M1pxMLx2">field of scalars</a>, it is called a <a href="/facts/Multilinear_form/Rk420AcK">multilinear form</a>. Multilinear maps and multilinear forms are fundamental objects of study in <a href="/facts/Multilinear_algebra/LrwpUFsM">multilinear algebra</a>.
If all variables belong to the same space, one can consider <a href="/facts/Symmetric_function/IB8N4T3m">symmetric</a>, <a href="/facts/Bilinear_form/gyvzn9ym">antisymmetric</a> and <a href="/facts/Alternating_map/ssrxVffJ">alternating</a> k-linear maps. The latter two coincide if the underlying <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> (or <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>) has a <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> different from two, else the former two coincide.

Multilinear map open-in-new

Multilinear map