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''anti-symmetric bilinear form''
 
''anti-symmetric bilinear form''
  
A [[Bilinear form|bilinear form]] $  f $
+
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
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 $  f $
+
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.
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 $  f $
+
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
on $  V $
 
there exists a basis $  e _ {1} \dots e _ {n} $
 
relative to which the matrix of $  f $
 
is of the form
 
  
$$ \tag{* }
+
<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>
\left \|
 
  
where $  m = w ( f  ) $
+
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.
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 $
+
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).
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 $
+
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
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
 
  
$$
+
<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>
0 \neq  f ( e _ {i} , e _ {i + m }  )  = \
 
\alpha _ {i}  \in  A,\ \
 
i = 1 \dots m,
 
$$
 
  
and $  \alpha _ {i} $
+
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" />.
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 $
+
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]]).
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====

Revision as of 14:53, 7 June 2020

anti-symmetric bilinear form

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

The structure of any skew-symmetric bilinear form on a finite-dimensional vector space over a field of characteristic is uniquely determined by its Witt index (see Witt theorem; Witt decomposition). Namely: is the orthogonal (with respect to ) direct sum of the kernel of and a subspace of dimension , the restriction of to which is a standard form. Two skew-symmetric bilinear forms on 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 is even.

For any skew-symmetric bilinear form on there exists a basis relative to which the matrix of is of the form

(*)

where and is the identity matrix of order . 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 over a field of characteristic there exists a non-singular matrix such that is of the form (*). In particular, the rank of 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 by the condition that the form be alternating: for any (for fields of characteristic the two conditions are equivalent).

These results can be generalized to the case where is a commutative principal ideal ring, is a free -module of finite dimension and is an alternating bilinear form on . To be precise: Under these assumptions there exists a basis of the module and a non-negative integer such that

and divides for ; otherwise . The ideals are uniquely determined by these conditions, and the module is generated by .

The determinant of an alternating matrix of odd order equals 0 for any commutative ring with an identity. In case the order of the alternating matrix over is even, the element is a square in (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