Editing Skew-symmetric matrix (section)

== Skew-symmetric and alternating forms ==

A '''skew-symmetric form''' <math>\varphi</math> on a [[vector space]] <math>V</math> over a [[field (mathematics)|field]] <math>K</math> of arbitrary characteristic is defined to be a [[bilinear form]]

<math display="block">\varphi: V \times V \mapsto K</math>

such that for all <math>v, w</math> in <math>V,</math>

<math display="block">\varphi(v, w) = -\varphi(w, v).</math>

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 is equivalent to that of a symmetric form, as every element is its own additive inverse.

Where the [[vector space]] <math>V</math> is over a field of arbitrary [[characteristic (algebra)|characteristic]] including characteristic 2, we may define an '''alternating form''' as a bilinear form <math>\varphi</math> such that for all vectors <math>v</math> in <math>V</math>

<math display="block">\varphi(v, v) = 0.</math>

This is equivalent to a skew-symmetric form when the field is not of characteristic 2, as seen from

<math display="block">0 = \varphi(v + w, v + w) = \varphi(v, v) + \varphi(v, w) + \varphi(w, v) + \varphi(w, w) = \varphi(v, w) + \varphi(w, v),</math>

whence

<math display="block">\varphi(v, w) = -\varphi(w, v).</math>

A bilinear form <math>\varphi</math> will be represented by a matrix <math>A</math> such that <math>\varphi(v,w) = v^\textsf{T}Aw</math>, once a [[basis (linear algebra)|basis]] of <math>V</math> is chosen, and conversely an <math>n \times n</math> matrix <math>A</math> on <math>K^n</math> gives rise to a form sending <math>(v, w)</math> to <math>v^\textsf{T}Aw.</math>  For each of symmetric, skew-symmetric and alternating forms, the representing matrices are symmetric, skew-symmetric and alternating respectively.