Skew-symmetric Matrix - Alternating Forms

Alternating Forms

We begin with a special case of the definition. An alternating form φ on a vector space V over a field K, not of characteristic 2, is defined to be a bilinear form

φ : V × V → K

such that

φ(v,w) = −φ(w,v).

This defines a form with desirable properties for vector spaces over fields of characteristic not equal to 2, but in a vector space over a field of characteristic 2, the definition fails, as every element is its own additive inverse. That is, symmetric and alternating forms are equivalent, which is clearly false in the case above. However, we may extend the definition to vector spaces over fields of characteristic 2 as follows:

In the case where the vector space V is over a field of arbitrary characteristic including characteristic 2, we may state that for all vectors v in V

φ(v,v) = 0.

This reduces to the above case when the field is not of characteristic 2 as seen below

0 = φ(v + w,v + w) = φ(v,v) + φ(v,w) + φ(w,v) + φ(w,w) = φ(v,w) + φ(w,v)

Whence,

φ(v,w) = −φ(w,v).

Thus, we have a definition that now holds for vector spaces over fields of all characteristics.

Such a φ will be represented by a skew-symmetric matrix A, φ(v, w) = vTAw, once a basis of V is chosen; and conversely an n×n skew-symmetric matrix A on Kn gives rise to an alternating form sending (v, w) to vTAw.