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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m0653002.png" />-linear mapping, multilinear operator''
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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m0653003.png" /> of the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m0653004.png" /> of unitary modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m0653005.png" /> (cf. [[Unitary module|Unitary module]]) over a commutative associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m0653006.png" /> with a unit into a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m0653007.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m0653008.png" /> which is linear in each argument, i.e. which satisfies the condition
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{{TEX|auto}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m0653009.png" /></td> </tr></table>
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'' $  n $-
 +
linear mapping, multilinear operator''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530010.png" /></td> </tr></table>
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A mapping  $  f $
 +
of the direct product  $  \prod _ {i=} 1  ^ {n} E _ {i} $
 +
of unitary modules  $  E _ {i} $(
 +
cf. [[Unitary module|Unitary module]]) over a commutative associative ring  $  A $
 +
with a unit into a certain  $  A $-
 +
module  $  F $
 +
which is linear in each argument, i.e. which satisfies the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530011.png" /></td> </tr></table>
+
$$
 +
f( x _ {1} \dots x _ {i-} 1 , ay + bz, x _ {i+} 1 \dots x _ {n} ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530012.png" /></td> </tr></table>
+
$$
 +
= \
 +
af( x _ {1} \dots x _ {i-} 1 , y, x _ {i+} 1 \dots x _ {n} ) +
 +
$$
  
In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530013.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530014.png" />) one speaks of a [[Bilinear mapping|bilinear mapping]] (respectively, a trilinear mapping). Each multilinear mapping
+
$$
 +
+
 +
bf ( x _ {i} \dots x _ {i-} 1 , z , x _ {i+} 1 \dots x _ {n} )
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530015.png" /></td> </tr></table>
+
$$
 +
( a, b  \in  A; \  y, z  \in  E _ {i} ,\  i  = 1 \dots n).
 +
$$
  
defines a unique linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530016.png" /> of the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530017.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530018.png" /> such that
+
In the case  $  n= 2 $(
 +
$  n= 3 $)
 +
one speaks of a [[Bilinear mapping|bilinear mapping]] (respectively, a trilinear mapping). Each multilinear mapping
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530019.png" /></td> </tr></table>
+
$$
 +
f: \prod _ { i= } 1 ^ { n }  E _ {i}  \rightarrow  F
 +
$$
  
where the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530020.png" /> is a bijection of the set of multilinear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530021.png" /> into the set of all linear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530022.png" />. The multilinear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530023.png" /> naturally form an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530024.png" />-module.
+
defines a unique linear mapping  $  \overline{f}\; $
 +
of the tensor product  $  \otimes _ {i=} 1  ^ {n} E _ {i} $
 +
into $  F $
 +
such that
  
On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530025.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530026.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530027.png" />-linear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530028.png" /> there acts the [[Symmetric group|symmetric group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530029.png" />:
+
$$
 +
\overline{f}\; ( x _ {1} \otimes \dots \otimes x _ {n} )  = \
 +
f( x _ {1} \dots x _ {n} ),\  x _ {i} \in E _ {i} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530030.png" /></td> </tr></table>
+
where the correspondence  $  f \mapsto \overline{f}\; $
 +
is a bijection of the set of multilinear mappings  $  \prod _ {i=} 1  ^ {n} E _ {i} \rightarrow F $
 +
into the set of all linear mappings  $  \otimes _ {i=} 1  ^ {n} E _ {i} \rightarrow F $.
 +
The multilinear mappings  $  \prod _ {i=} 1  ^ {n} E _ {i} \rightarrow F $
 +
naturally form an  $  A $-
 +
module.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530033.png" />. A multilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530034.png" /> is called symmetric if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530035.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530036.png" />, and skew-symmetric if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530038.png" /> in accordance with the sign of the permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530039.png" />. A multilinear mapping is called sign-varying (or alternating) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530040.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530041.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530042.png" />. Any alternating multilinear mapping is skew-symmetric, while if in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530043.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530044.png" /> has the unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530045.png" /> the converse also holds. The symmetric multilinear mappings form a submodule in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530046.png" /> that is naturally isomorphic to the module of linear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530047.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530048.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530049.png" />-th symmetric power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530050.png" /> (see [[Symmetric algebra|Symmetric algebra]]). The alternating multilinear mappings form a submodule that is naturally isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530051.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530052.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530053.png" />-th exterior power of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530054.png" /> (see [[Exterior algebra|Exterior algebra]]). The multilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530055.png" /> is called the symmetrized multilinear mapping defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530056.png" />, while the multilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530057.png" /> is called the skew-symmetrized mapping defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530058.png" />. Symmetrized (skew-symmetrized) multilinear mappings are symmetric (respectively, alternating), and if in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530059.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530060.png" /> has a unique solution for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530061.png" />, then the converse is true. A sufficient condition for any alternating multilinear mapping to be a skew-symmetrization is that the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065300/m06530062.png" /> is free (cf. [[Free module|Free module]]). For references see [[Multilinear form|Multilinear form]].
+
On the  $  A $-
 +
module  $  L _ {n} ( E, F  ) $
 +
of all  $  n $-
 +
linear mappings  $  E  ^ {n} \rightarrow F $
 +
there acts the [[Symmetric group|symmetric group]]  $  S _ {n} $:
 +
 
 +
$$
 +
( sf  )( x _ {1} \dots x _ {n} )  = \
 +
f( x _ {s(} 1) \dots x _ {s(} n) ),
 +
$$
 +
 
 +
where $  s \in S _ {n} $,
 +
$  f \in L _ {n} ( E, F  ) $,  
 +
$  x _ {i} \in E $.  
 +
A multilinear mapping $  f $
 +
is called symmetric if $  sf = f $
 +
for all $  s \in S _ {n} $,  
 +
and skew-symmetric if $  sf = \epsilon ( s) f $,  
 +
where $  \epsilon ( s) = \pm  1 $
 +
in accordance with the sign of the permutation $  s $.  
 +
A multilinear mapping is called sign-varying (or alternating) if $  f( x _ {1} \dots x _ {n} ) = 0 $
 +
when $  x _ {i} = x _ {j} $
 +
for some $  i \neq j $.  
 +
Any alternating multilinear mapping is skew-symmetric, while if in $  F $
 +
the equation $  2y = 0 $
 +
has the unique solution $  y = 0 $
 +
the converse also holds. The symmetric multilinear mappings form a submodule in $  L _ {n} ( E, F  ) $
 +
that is naturally isomorphic to the module of linear mappings $  L( S  ^ {n} E, F  ) $,  
 +
where $  S  ^ {n} E $
 +
is the $  n $-
 +
th symmetric power of $  E $(
 +
see [[Symmetric algebra|Symmetric algebra]]). The alternating multilinear mappings form a submodule that is naturally isomorphic to $  L( \Lambda  ^ {n} E, F  ) $,  
 +
where $  \Lambda  ^ {n} E $
 +
is the $  n $-
 +
th exterior power of the module $  E $(
 +
see [[Exterior algebra|Exterior algebra]]). The multilinear mapping $  \alpha _ {n} f = \sum _ {s \in S _ {n}  } sf $
 +
is called the symmetrized multilinear mapping defined by $  f $,  
 +
while the multilinear mapping $  \sigma _ {n} f = \sum _ {s \in S _ {n}  } \epsilon ( s) sf $
 +
is called the skew-symmetrized mapping defined by $  f $.  
 +
Symmetrized (skew-symmetrized) multilinear mappings are symmetric (respectively, alternating), and if in $  F $
 +
the equation $  n!y = c $
 +
has a unique solution for each $  c \in F $,  
 +
then the converse is true. A sufficient condition for any alternating multilinear mapping to be a skew-symmetrization is that the module $  E $
 +
is free (cf. [[Free module|Free module]]). For references see [[Multilinear form|Multilinear form]].

Latest revision as of 08:02, 6 June 2020


$ n $- linear mapping, multilinear operator

A mapping $ f $ of the direct product $ \prod _ {i=} 1 ^ {n} E _ {i} $ of unitary modules $ E _ {i} $( cf. Unitary module) over a commutative associative ring $ A $ with a unit into a certain $ A $- module $ F $ which is linear in each argument, i.e. which satisfies the condition

$$ f( x _ {1} \dots x _ {i-} 1 , ay + bz, x _ {i+} 1 \dots x _ {n} ) = $$

$$ = \ af( x _ {1} \dots x _ {i-} 1 , y, x _ {i+} 1 \dots x _ {n} ) + $$

$$ + bf ( x _ {i} \dots x _ {i-} 1 , z , x _ {i+} 1 \dots x _ {n} ) $$

$$ ( a, b \in A; \ y, z \in E _ {i} ,\ i = 1 \dots n). $$

In the case $ n= 2 $( $ n= 3 $) one speaks of a bilinear mapping (respectively, a trilinear mapping). Each multilinear mapping

$$ f: \prod _ { i= } 1 ^ { n } E _ {i} \rightarrow F $$

defines a unique linear mapping $ \overline{f}\; $ of the tensor product $ \otimes _ {i=} 1 ^ {n} E _ {i} $ into $ F $ such that

$$ \overline{f}\; ( x _ {1} \otimes \dots \otimes x _ {n} ) = \ f( x _ {1} \dots x _ {n} ),\ x _ {i} \in E _ {i} , $$

where the correspondence $ f \mapsto \overline{f}\; $ is a bijection of the set of multilinear mappings $ \prod _ {i=} 1 ^ {n} E _ {i} \rightarrow F $ into the set of all linear mappings $ \otimes _ {i=} 1 ^ {n} E _ {i} \rightarrow F $. The multilinear mappings $ \prod _ {i=} 1 ^ {n} E _ {i} \rightarrow F $ naturally form an $ A $- module.

On the $ A $- module $ L _ {n} ( E, F ) $ of all $ n $- linear mappings $ E ^ {n} \rightarrow F $ there acts the symmetric group $ S _ {n} $:

$$ ( sf )( x _ {1} \dots x _ {n} ) = \ f( x _ {s(} 1) \dots x _ {s(} n) ), $$

where $ s \in S _ {n} $, $ f \in L _ {n} ( E, F ) $, $ x _ {i} \in E $. A multilinear mapping $ f $ is called symmetric if $ sf = f $ for all $ s \in S _ {n} $, and skew-symmetric if $ sf = \epsilon ( s) f $, where $ \epsilon ( s) = \pm 1 $ in accordance with the sign of the permutation $ s $. A multilinear mapping is called sign-varying (or alternating) if $ f( x _ {1} \dots x _ {n} ) = 0 $ when $ x _ {i} = x _ {j} $ for some $ i \neq j $. Any alternating multilinear mapping is skew-symmetric, while if in $ F $ the equation $ 2y = 0 $ has the unique solution $ y = 0 $ the converse also holds. The symmetric multilinear mappings form a submodule in $ L _ {n} ( E, F ) $ that is naturally isomorphic to the module of linear mappings $ L( S ^ {n} E, F ) $, where $ S ^ {n} E $ is the $ n $- th symmetric power of $ E $( see Symmetric algebra). The alternating multilinear mappings form a submodule that is naturally isomorphic to $ L( \Lambda ^ {n} E, F ) $, where $ \Lambda ^ {n} E $ is the $ n $- th exterior power of the module $ E $( see Exterior algebra). The multilinear mapping $ \alpha _ {n} f = \sum _ {s \in S _ {n} } sf $ is called the symmetrized multilinear mapping defined by $ f $, while the multilinear mapping $ \sigma _ {n} f = \sum _ {s \in S _ {n} } \epsilon ( s) sf $ is called the skew-symmetrized mapping defined by $ f $. Symmetrized (skew-symmetrized) multilinear mappings are symmetric (respectively, alternating), and if in $ F $ the equation $ n!y = c $ has a unique solution for each $ c \in F $, then the converse is true. A sufficient condition for any alternating multilinear mapping to be a skew-symmetrization is that the module $ E $ is free (cf. Free module). For references see Multilinear form.

How to Cite This Entry:
Multilinear mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multilinear_mapping&oldid=15130
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article