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''anti-symmetric bilinear form''
 
''anti-symmetric bilinear form''
  
A [[Bilinear form|bilinear form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s0857101.png" /> on a unitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s0857102.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s0857103.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s0857104.png" /> is a commutative ring with an identity) such that
+
A [[Bilinear form|bilinear form]] $  f $
 +
on a unitary $  A $-
 +
module $  V $(
 +
where $  A $
 +
is a commutative ring with an identity) such that
  
<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/s/s085/s085710/s0857105.png" /></td> </tr></table>
+
$$
 +
f ( v _ {1} , v _ {2} )  = \
 +
- f ( v _ {2} , v _ {1} ) \ \
 +
\textrm{ for }  \textrm{ all } \
 +
v _ {1} , v _ {2} \in V.
 +
$$
  
The structure of any skew-symmetric bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s0857106.png" /> on a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s0857107.png" /> over a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s0857108.png" /> is uniquely determined by its Witt index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s0857109.png" /> (see [[Witt theorem|Witt theorem]]; [[Witt decomposition|Witt decomposition]]). Namely: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571010.png" /> is the orthogonal (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571011.png" />) direct sum of the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571013.png" /> and a subspace of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571014.png" />, the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571015.png" /> to which is a standard form. Two skew-symmetric bilinear forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571016.png" /> are isometric if and only if their Witt indices are equal. In particular, a non-degenerate skew-symmetric bilinear form is standard, and in that case the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571017.png" /> is even.
+
The structure of any skew-symmetric bilinear form $  f $
 +
on a finite-dimensional vector space $  V $
 +
over a field of characteristic $  \neq 2 $
 +
is uniquely determined by its Witt index $  w ( f  ) $(
 +
see [[Witt theorem|Witt theorem]]; [[Witt decomposition|Witt decomposition]]). Namely: $  V $
 +
is the orthogonal (with respect to $  f  $)  
 +
direct sum of the kernel $  V  ^  \perp  $
 +
of $  f $
 +
and a subspace of dimension $  2w ( f  ) $,  
 +
the restriction of $  f $
 +
to which is a standard form. Two skew-symmetric bilinear forms on $  V $
 +
are isometric if and only if their Witt indices are equal. In particular, a non-degenerate skew-symmetric bilinear form is standard, and in that case the dimension of $  V $
 +
is even.
  
For any skew-symmetric bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571019.png" /> there exists a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571020.png" /> relative to which the matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571021.png" /> is of the form
+
For any skew-symmetric bilinear form $  f $
 +
on $  V $
 +
there exists a basis $  e _ {1}, \dots, e _ {n} $
 +
relative to which the matrix of $  f $
 +
is of 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/s/s085/s085710/s08571022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\left \|
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571024.png" /> is the identity matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571025.png" />. The matrix of a skew-symmetric bilinear form relative to any basis is skew-symmetric. Therefore, the above properties of skew-symmetric bilinear forms can be formulated as follows: For any skew-symmetric matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571026.png" /> over a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571027.png" /> there exists a non-singular matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571029.png" /> is of the form (*). In particular, the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571030.png" /> is even, and the determinant of a skew-symmetric matrix of odd order is 0.
+
\begin{array}{ccc}
 +
0 &E _ {m}  & 0 \\
 +
- E _ {m}  & 0 & 0 \\
 +
0 & 0 & 0 \\
 +
\end{array}
 +
\
 +
\right \| ,
 +
$$
  
The above assertions remain valid for a field of characteristic 2, provided one replaces the skew-symmetry condition for the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571031.png" /> by the condition that the form be alternating: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571032.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571033.png" /> (for fields of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571034.png" /> the two conditions are equivalent).
+
where  $  m = w ( f  ) $
 +
and  $  E _ {m} $
 +
is the identity matrix of order  $  m $.
 +
The matrix of a skew-symmetric bilinear form relative to any basis is skew-symmetric. Therefore, the above properties of skew-symmetric bilinear forms can be formulated as follows: For any skew-symmetric matrix  $  M $
 +
over a field of characteristic $  \neq 2 $
 +
there exists a non-singular matrix  $  P $
 +
such that  $  P  ^ {T} MP $
 +
is of the form (*). In particular, the rank of  $  M $
 +
is even, and the determinant of a skew-symmetric matrix of odd order is 0.
  
These results can be generalized to the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571035.png" /> is a commutative principal ideal ring, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571036.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571037.png" />-module of finite dimension and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571038.png" /> is an alternating bilinear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571039.png" />. To be precise: Under these assumptions there exists a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571040.png" /> of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571041.png" /> and a non-negative integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571042.png" /> such that
+
The above assertions remain valid for a field of characteristic 2, provided one replaces the skew-symmetry condition for the form  $  f $
 +
by the condition that the form be alternating: $  f ( v, v) = 0 $
 +
for any  $  v \in V $ (for fields of characteristic  $  \neq 2 $
 +
the two conditions are equivalent).
  
<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/s/s085/s085710/s08571043.png" /></td> </tr></table>
+
These results can be generalized to the case where  $  A $
 +
is a commutative principal ideal ring,  $  V $
 +
is a free  $  A $-
 +
module of finite dimension and  $  f $
 +
is an alternating bilinear form on  $  V $.
 +
To be precise: Under these assumptions there exists a basis  $  e _ {1}, \dots, e _ {n} $
 +
of the module  $  V $
 +
and a non-negative integer  $  m \leq  n/2 $
 +
such that
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571044.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571045.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571046.png" />; otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571047.png" />. The ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571048.png" /> are uniquely determined by these conditions, and the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571049.png" /> is generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571050.png" />.
+
$$
 +
0 \neq  f ( e _ {i} , e _ {i + m }  )  = \
 +
\alpha _ {i}  \in  A,\ \
 +
i = 1, \dots, m,
 +
$$
  
The determinant of an alternating matrix of odd order equals 0 for any commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571051.png" /> with an identity. In case the order of the alternating matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571052.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571053.png" /> is even, the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571054.png" /> is a square in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085710/s08571055.png" /> (see [[Pfaffian|Pfaffian]]).
+
and  $  \alpha _ {i} $
 +
divides  $  \alpha _ {i + 1 }  $
 +
for  $  i = 1, \dots, m - 1 $;
 +
otherwise  $  f ( e _ {i} , e _ {j} ) = 0 $.
 +
The ideals  $  A \alpha _ {i} $
 +
are uniquely determined by these conditions, and the module  $  V  ^  \perp  $
 +
is generated by  $  e _ {2m + 1 }, \dots, e _ {n} $.
 +
 
 +
The determinant of an alternating matrix of odd order equals 0 for any commutative ring $  A $
 +
with an identity. In case the order of the alternating matrix $  M $
 +
over $  A $
 +
is even, the element $  \mathop{\rm det}  M \in A $
 +
is a square in $  A $ (see [[Pfaffian|Pfaffian]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Algèbre" , ''Eléments de mathématiques'' , Hermann  (1970)  pp. Chapt. II. Algèbre linéaire</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1984)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Artin,  "Geometric algebra" , Interscience  (1957)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Algèbre" , ''Eléments de mathématiques'' , Hermann  (1970)  pp. Chapt. II. Algèbre linéaire</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1984)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Artin,  "Geometric algebra" , Interscience  (1957)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 03:29, 21 March 2022


anti-symmetric bilinear form

A bilinear form $ f $ on a unitary $ A $- module $ V $( where $ A $ is a commutative ring with an identity) such that

$$ f ( v _ {1} , v _ {2} ) = \ - f ( v _ {2} , v _ {1} ) \ \ \textrm{ for } \textrm{ all } \ v _ {1} , v _ {2} \in V. $$

The structure of any skew-symmetric bilinear form $ f $ on a finite-dimensional vector space $ V $ over a field of characteristic $ \neq 2 $ is uniquely determined by its Witt index $ w ( f ) $( see Witt theorem; Witt decomposition). Namely: $ V $ is the orthogonal (with respect to $ f $) direct sum of the kernel $ V ^ \perp $ of $ f $ and a subspace of dimension $ 2w ( f ) $, the restriction of $ f $ to which is a standard form. Two skew-symmetric bilinear forms on $ V $ are isometric if and only if their Witt indices are equal. In particular, a non-degenerate skew-symmetric bilinear form is standard, and in that case the dimension of $ V $ is even.

For any skew-symmetric bilinear form $ f $ on $ V $ there exists a basis $ e _ {1}, \dots, e _ {n} $ relative to which the matrix of $ f $ is of the form

$$ \tag{* } \left \| \begin{array}{ccc} 0 &E _ {m} & 0 \\ - E _ {m} & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \ \right \| , $$

where $ m = w ( f ) $ and $ E _ {m} $ is the identity matrix of order $ m $. The matrix of a skew-symmetric bilinear form relative to any basis is skew-symmetric. Therefore, the above properties of skew-symmetric bilinear forms can be formulated as follows: For any skew-symmetric matrix $ M $ over a field of characteristic $ \neq 2 $ there exists a non-singular matrix $ P $ such that $ P ^ {T} MP $ is of the form (*). In particular, the rank of $ M $ is even, and the determinant of a skew-symmetric matrix of odd order is 0.

The above assertions remain valid for a field of characteristic 2, provided one replaces the skew-symmetry condition for the form $ f $ by the condition that the form be alternating: $ f ( v, v) = 0 $ for any $ v \in V $ (for fields of characteristic $ \neq 2 $ the two conditions are equivalent).

These results can be generalized to the case where $ A $ is a commutative principal ideal ring, $ V $ is a free $ A $- module of finite dimension and $ f $ is an alternating bilinear form on $ V $. To be precise: Under these assumptions there exists a basis $ e _ {1}, \dots, e _ {n} $ of the module $ V $ and a non-negative integer $ m \leq n/2 $ such that

$$ 0 \neq f ( e _ {i} , e _ {i + m } ) = \ \alpha _ {i} \in A,\ \ i = 1, \dots, m, $$

and $ \alpha _ {i} $ divides $ \alpha _ {i + 1 } $ for $ i = 1, \dots, m - 1 $; otherwise $ f ( e _ {i} , e _ {j} ) = 0 $. The ideals $ A \alpha _ {i} $ are uniquely determined by these conditions, and the module $ V ^ \perp $ is generated by $ e _ {2m + 1 }, \dots, e _ {n} $.

The determinant of an alternating matrix of odd order equals 0 for any commutative ring $ A $ with an identity. In case the order of the alternating matrix $ M $ over $ A $ is even, the element $ \mathop{\rm det} M \in A $ is a square in $ A $ (see Pfaffian).

References

[1] N. Bourbaki, "Algèbre" , Eléments de mathématiques , Hermann (1970) pp. Chapt. II. Algèbre linéaire
[2] S. Lang, "Algebra" , Addison-Wesley (1984)
[3] E. Artin, "Geometric algebra" , Interscience (1957)

Comments

The kernel of a skew-symmetric bilinear form is the left kernel of the corresponding bilinear mapping, which is equal to the right kernel by skew symmetry.

References

[a1] J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973)
How to Cite This Entry:
Skew-symmetric bilinear form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Skew-symmetric_bilinear_form&oldid=13596
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article