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Difference between revisions of "Skew-symmetric bilinear form"

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For any skew-symmetric bilinear form  $  f $
 
For any skew-symmetric bilinear form  $  f $
 
on  $  V $
 
on  $  V $
there exists a basis  $  e _ {1} \dots e _ {n} $
+
there exists a basis  $  e _ {1}, \dots, e _ {n} $
 
relative to which the matrix of  $  f $
 
relative to which the matrix of  $  f $
 
is of the form
 
is of the form
Line 48: Line 48:
 
$$ \tag{* }
 
$$ \tag{* }
 
\left \|
 
\left \|
 +
 +
\begin{array}{ccc}
 +
0  &E _ {m}  & 0  \\
 +
- E _ {m}  & 0  & 0  \\
 +
0  & 0  & 0  \\
 +
\end{array}
 +
\
 +
\right \| ,
 +
$$
  
 
where  $  m = w ( f  ) $
 
where  $  m = w ( f  ) $
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The above assertions remain valid for a field of characteristic 2, provided one replaces the skew-symmetry condition for the form  $  f $
 
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 $
 
by the condition that the form be alternating:  $  f ( v, v) = 0 $
for any  $  v \in V $(
+
for any  $  v \in V $ (for fields of characteristic  $  \neq 2 $
for fields of characteristic  $  \neq 2 $
 
 
the two conditions are equivalent).
 
the two conditions are equivalent).
  
Line 70: Line 78:
 
module of finite dimension and  $  f $
 
module of finite dimension and  $  f $
 
is an alternating bilinear form on  $  V $.  
 
is an alternating bilinear form on  $  V $.  
To be precise: Under these assumptions there exists a basis  $  e _ {1} \dots e _ {n} $
+
To be precise: Under these assumptions there exists a basis  $  e _ {1}, \dots, e _ {n} $
 
of the module  $  V $
 
of the module  $  V $
 
and a non-negative integer  $  m \leq  n/2 $
 
and a non-negative integer  $  m \leq  n/2 $
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0  \neq  f ( e _ {i} , e _ {i + m }  )  = \  
 
0  \neq  f ( e _ {i} , e _ {i + m }  )  = \  
 
\alpha _ {i}  \in  A,\ \  
 
\alpha _ {i}  \in  A,\ \  
i = 1 \dots m,
+
i = 1, \dots, m,
 
$$
 
$$
  
 
and  $  \alpha _ {i} $
 
and  $  \alpha _ {i} $
 
divides  $  \alpha _ {i + 1 }  $
 
divides  $  \alpha _ {i + 1 }  $
for  $  i = 1 \dots m - 1 $;  
+
for  $  i = 1, \dots, m - 1 $;  
 
otherwise  $  f ( e _ {i} , e _ {j} ) = 0 $.  
 
otherwise  $  f ( e _ {i} , e _ {j} ) = 0 $.  
 
The ideals  $  A \alpha _ {i} $
 
The ideals  $  A \alpha _ {i} $
 
are uniquely determined by these conditions, and the module  $  V  ^  \perp  $
 
are uniquely determined by these conditions, and the module  $  V  ^  \perp  $
is generated by  $  e _ {2m + 1 } \dots e _ {n} $.
+
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 $
 
The determinant of an alternating matrix of odd order equals 0 for any commutative ring  $  A $
Line 93: Line 101:
 
over  $  A $
 
over  $  A $
 
is even, the element  $  \mathop{\rm det}  M \in A $
 
is even, the element  $  \mathop{\rm det}  M \in A $
is a square in  $  A $(
+
is a square in  $  A $ (see [[Pfaffian|Pfaffian]]).
see [[Pfaffian|Pfaffian]]).
 
  
 
====References====
 
====References====

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=48725
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article