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A fundamental operation in the [[Exterior algebra|exterior algebra]] of tensors defined on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e0371301.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e0371302.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e0371303.png" />.
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$#A+1 = 58 n = 0
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$#C+1 = 58 : ~/encyclopedia/old_files/data/E037/E.0307130 Exterior product
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Automatically converted into TeX, above some diagnostics.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e0371304.png" /> be a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e0371305.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e0371306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e0371307.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e0371308.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e0371309.png" />-forms:
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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/e/e037/e037130/e03713010.png" /></td> </tr></table>
+
A fundamental operation in the [[Exterior algebra|exterior algebra]] of tensors defined on an  $  n $-
 +
dimensional vector space  $  V $
 +
over a field  $  K $.
  
<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/e/e037/e037130/e03713011.png" /></td> </tr></table>
+
Let  $  e _ {1} \dots e _ {n} $
 +
be a basis of  $  V $,
 +
and let  $  a $
 +
and  $  b $
 +
be  $  p $-
 +
and  $  q $-
 +
forms:
  
The exterior product of the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713013.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713014.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713015.png" /> obtained by [[Alternation|alternation]] of the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713016.png" />. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713017.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713018.png" />; its coordinates are skew-symmetric:
+
$$
 +
= a ^ {i _ {1} \dots {i _ {p} } }
 +
e _ {i _ {1}  }  \otimes \dots \otimes  e _ {i _ {p}  } ,
 +
$$
  
<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/e/e037/e037130/e03713019.png" /></td> </tr></table>
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$$
 +
= b ^ {j _ {1} \dots {j _ {q} } } e _ {j _ {1}  }  \otimes \dots \otimes  e _ {j _ {q}  } .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713020.png" /> are the components of the generalized [[Kronecker symbol|Kronecker symbol]]. The exterior product of covariant tensors is defined in a similar manner.
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The exterior product of the forms  $  a $
 +
and  $  b $
 +
is the  $  ( p + q) $-
 +
form  $  c $
 +
obtained by [[Alternation|alternation]] of the tensor product  $  a \otimes b $.
 +
The form  $  c $
 +
is denoted by  $  a \wedge b $;
 +
its coordinates are skew-symmetric:
 +
 
 +
$$
 +
c ^ {k _ {1} \dots k _ {p+ q }  }  = \
 +
 
 +
\frac{1}{p! q! }
 +
 
 +
\delta _ {i _ {1}  \dots i _ {p} j _ {1} \dots j _ {q} } ^ {k _ {1} \dots \dots \dots k _ {p+ q }  }
 +
a ^ {i _ {1} \dots i _ {p} } b ^ {j _ {1} \dots j _ {q} } ,
 +
$$
 +
 
 +
where $  \delta _ {i _ {1}  \dots j _ {q} } ^ {k _ {1} \dots k _ {p+} q } $
 +
are the components of the generalized [[Kronecker symbol|Kronecker symbol]]. The exterior product of covariant tensors is defined in a similar manner.
  
 
The basic properties of the exterior product are listed below:
 
The basic properties of the exterior product are listed below:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713022.png" /> (homogeneity);
+
1) $  ( ka) \wedge b = a \wedge ( kb) = k( a \wedge b) $,  
 +
$  k \in K $(
 +
homogeneity);
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713023.png" /> (distributivity);
+
2) $  ( a+ b) \wedge c = a \wedge c + b \wedge c $(
 +
distributivity);
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713024.png" /> (associativity).
+
3) $  ( a \wedge b ) \wedge c = a \wedge ( b \wedge c) $(
 +
associativity).
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713025.png" />; if the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713026.png" /> is distinct from two, the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713027.png" /> is valid for any form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713028.png" /> of odd valency.
+
4) $  a \wedge b = (- 1)  ^ {pq} b \wedge a $;  
 +
if the characteristic of $  K $
 +
is distinct from two, the equation $  a \wedge a = 0 $
 +
is valid for any form $  a $
 +
of odd valency.
  
The exterior product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713029.png" /> vectors is said to be a decomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713031.png" />-vector. Any [[Poly-vector|poly-vector]] of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713032.png" /> is a linear combination of decomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713033.png" />-vectors. The components of this combination are the (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713034.png" />)-minors of the (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713035.png" />)-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713038.png" />, of the coefficients of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713040.png" /> their exterior product has the form
+
The exterior product of $  s $
 +
vectors is said to be a decomposable $  s $-
 +
vector. Any [[Poly-vector|poly-vector]] of dimension $  s $
 +
is a linear combination of decomposable $  s $-
 +
vectors. The components of this combination are the ( $  s \times s $)-
 +
minors of the ( $  n \times s $)-
 +
matrix $  ( a _ {j}  ^ {i} ) $,  
 +
$  1 \leq  i \leq  n $,  
 +
$  1 \leq  j \leq  s $,  
 +
of the coefficients of the vectors $  a _ {1} \dots a _ {s} $.  
 +
If $  s = n $
 +
their exterior product has the form
  
<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/e/e037/e037130/e03713041.png" /></td> </tr></table>
+
$$
 +
\alpha _ {n}  = a _ {1} \wedge \dots \wedge a _ {n}  = \
 +
\mathop{\rm det} ( a _ {j}  ^ {i} )  e _ {1} \wedge \dots \wedge e _ {n} .
 +
$$
  
Over fields of characteristic distinct from two, the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713042.png" /> is necessary and sufficient for vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713043.png" /> to be linearly dependent. A non-zero decomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713044.png" />-vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713045.png" /> defines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713046.png" /> an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713047.png" />-dimensional oriented subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713048.png" />, parallel to the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713049.png" />, and the [[Parallelotope|parallelotope]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713050.png" /> formed by the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713051.png" /> issuing from one point, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713052.png" />. The conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713054.png" /> are equivalent.
+
Over fields of characteristic distinct from two, the equation $  a _ {1} \wedge \dots \wedge a _ {n} = 0 $
 +
is necessary and sufficient for vectors $  a _ {1} \dots a _ {n} $
 +
to be linearly dependent. A non-zero decomposable $  s $-
 +
vector $  \alpha _ {s} $
 +
defines in $  V $
 +
an $  s $-
 +
dimensional oriented subspace $  A $,  
 +
parallel to the vectors $  a _ {1} \dots a _ {s} $,  
 +
and the [[Parallelotope|parallelotope]] in $  A $
 +
formed by the vectors $  a _ {1} \dots a _ {s} $
 +
issuing from one point, denoted by $  [ a _ {1} \dots a _ {s} ] $.  
 +
The conditions $  a \in A $
 +
and $  \alpha _ {s} \wedge a = 0 $
 +
are equivalent.
  
 
For references see [[Exterior algebra|Exterior algebra]].
 
For references see [[Exterior algebra|Exterior algebra]].
 
 
  
 
====Comments====
 
====Comments====
Instead of exterior product the phrase  "outer product"  is sometimes used. The condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713055.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713056.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713058.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037130/e03713059.png" /> is sometimes called graded commutativity.
+
Instead of exterior product the phrase  "outer product"  is sometimes used. The condition $  a \wedge b = (- 1)  ^ {pq} b \wedge a $
 +
for $  a $
 +
of degree $  p $
 +
and $  b $
 +
of degree $  q $
 +
is sometimes called graded commutativity.

Latest revision as of 19:38, 5 June 2020


A fundamental operation in the exterior algebra of tensors defined on an $ n $- dimensional vector space $ V $ over a field $ K $.

Let $ e _ {1} \dots e _ {n} $ be a basis of $ V $, and let $ a $ and $ b $ be $ p $- and $ q $- forms:

$$ a = a ^ {i _ {1} \dots {i _ {p} } } e _ {i _ {1} } \otimes \dots \otimes e _ {i _ {p} } , $$

$$ b = b ^ {j _ {1} \dots {j _ {q} } } e _ {j _ {1} } \otimes \dots \otimes e _ {j _ {q} } . $$

The exterior product of the forms $ a $ and $ b $ is the $ ( p + q) $- form $ c $ obtained by alternation of the tensor product $ a \otimes b $. The form $ c $ is denoted by $ a \wedge b $; its coordinates are skew-symmetric:

$$ c ^ {k _ {1} \dots k _ {p+ q } } = \ \frac{1}{p! q! } \delta _ {i _ {1} \dots i _ {p} j _ {1} \dots j _ {q} } ^ {k _ {1} \dots \dots \dots k _ {p+ q } } a ^ {i _ {1} \dots i _ {p} } b ^ {j _ {1} \dots j _ {q} } , $$

where $ \delta _ {i _ {1} \dots j _ {q} } ^ {k _ {1} \dots k _ {p+} q } $ are the components of the generalized Kronecker symbol. The exterior product of covariant tensors is defined in a similar manner.

The basic properties of the exterior product are listed below:

1) $ ( ka) \wedge b = a \wedge ( kb) = k( a \wedge b) $, $ k \in K $( homogeneity);

2) $ ( a+ b) \wedge c = a \wedge c + b \wedge c $( distributivity);

3) $ ( a \wedge b ) \wedge c = a \wedge ( b \wedge c) $( associativity).

4) $ a \wedge b = (- 1) ^ {pq} b \wedge a $; if the characteristic of $ K $ is distinct from two, the equation $ a \wedge a = 0 $ is valid for any form $ a $ of odd valency.

The exterior product of $ s $ vectors is said to be a decomposable $ s $- vector. Any poly-vector of dimension $ s $ is a linear combination of decomposable $ s $- vectors. The components of this combination are the ( $ s \times s $)- minors of the ( $ n \times s $)- matrix $ ( a _ {j} ^ {i} ) $, $ 1 \leq i \leq n $, $ 1 \leq j \leq s $, of the coefficients of the vectors $ a _ {1} \dots a _ {s} $. If $ s = n $ their exterior product has the form

$$ \alpha _ {n} = a _ {1} \wedge \dots \wedge a _ {n} = \ \mathop{\rm det} ( a _ {j} ^ {i} ) e _ {1} \wedge \dots \wedge e _ {n} . $$

Over fields of characteristic distinct from two, the equation $ a _ {1} \wedge \dots \wedge a _ {n} = 0 $ is necessary and sufficient for vectors $ a _ {1} \dots a _ {n} $ to be linearly dependent. A non-zero decomposable $ s $- vector $ \alpha _ {s} $ defines in $ V $ an $ s $- dimensional oriented subspace $ A $, parallel to the vectors $ a _ {1} \dots a _ {s} $, and the parallelotope in $ A $ formed by the vectors $ a _ {1} \dots a _ {s} $ issuing from one point, denoted by $ [ a _ {1} \dots a _ {s} ] $. The conditions $ a \in A $ and $ \alpha _ {s} \wedge a = 0 $ are equivalent.

For references see Exterior algebra.

Comments

Instead of exterior product the phrase "outer product" is sometimes used. The condition $ a \wedge b = (- 1) ^ {pq} b \wedge a $ for $ a $ of degree $ p $ and $ b $ of degree $ q $ is sometimes called graded commutativity.

How to Cite This Entry:
Exterior product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exterior_product&oldid=46889
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article