Skew-symmetric bilinear form

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)