alternating form

A bilinear formPlanetmathPlanetmath A on a vector spaceMathworldPlanetmath V (over a field k) is called an alternating form if for all vV, A(v,v)=0.

Since for any u,vV,


we see that A(u,v)=-A(v,u). So an alternating form is automatically a anti-symmetric, or skew symmetric form. The converse is true if the characteristic of k is not 2.

Let V be a two dimensional vector space over k with an alternating form A. Let {e1,e2} be a basis for V. The matrix associated with A looks like


where r=A(e1,e2). The skew symmetric matrix S has the property that its diagonal entries are all 0. S is called the 2×2 alternating or symplectic matrix.

A is called non-singular or non-degenerate if there exist a vectors u,vV such that A(u,v)0. u,v are necessarily non-zero. Note that the associated matrix rS is non-singular iff r0 iff A is non-singular.

In the two dimensional vector space case above, if A is non-singular, we can re-scale the basis elements so that r=1. This means that the matrix associated with A is the alternating matrix. A two-dimensional vector space which carries a non-singular alternating form is sometimes called an alternating or symplectic hyperbolic plane. Some authors also call it simply a hyperbolic plane. But here on PlanetMath, we will reserve the shorter name for its cousin in the category of quadratic spaces. Let’s denote an alternating hyperbolic plane by 𝒜.

Remark. In general, it can be shown that if V is an n-dimensional vector space equipped with a non-singular alternating form A, then V can be written as an orthogonal direct sum of the alternating hyperbolic planes 𝒜. In other words, the associated matrix for A has the block form

(S𝟎𝟎𝟎S𝟎𝟎𝟎S), where 𝟎=(0000).

Furthermore, n is even. V is called a symplectic vector space.

Title alternating form
Canonical name AlternatingForm
Date of creation 2013-03-22 15:42:17
Last modified on 2013-03-22 15:42:17
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 15A63
Synonym alternate form
Synonym alternating
Synonym symplectic hyperbolic plane
Related topic SymplecticVectorSpace
Related topic EverySymplecticManifoldHasEvenDimension
Defines alternating hyperbolic plane