Since for any ,
Let be a two dimensional vector space over with an alternating form . Let be a basis for . The matrix associated with looks like
is called non-singular or non-degenerate if there exist a vectors such that . are necessarily non-zero. Note that the associated matrix is non-singular iff iff is non-singular.
In the two dimensional vector space case above, if is non-singular, we can re-scale the basis elements so that . This means that the matrix associated with 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 .
Remark. In general, it can be shown that if is an -dimensional vector space equipped with a non-singular alternating form , then can be written as an orthogonal direct sum of the alternating hyperbolic planes . In other words, the associated matrix for has the block form
Furthermore, is even. is called a symplectic vector space.
|Date of creation||2013-03-22 15:42:17|
|Last modified on||2013-03-22 15:42:17|
|Last modified by||CWoo (3771)|
|Synonym||symplectic hyperbolic plane|
|Defines||alternating hyperbolic plane|