# alternating form

A bilinear form^{} $A$ on a vector space^{} $V$ (over a field $k$) is called an *alternating form* if for all $v\in V$, $A(v,v)=0$.

Since for any $u,v\in V$,

$$0=A(u+v,u+v)=A(u,u)+A(u,v)+A(v,u)+A(v,v)=A(u,v)+A(v,u),$$ |

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 $\{{e}_{1},{e}_{2}\}$ be a basis for $V$. The matrix associated with $A$ looks like

$\left(\begin{array}{cc}\hfill A({e}_{1},{e}_{1})\hfill & \hfill A({e}_{1},{e}_{2})\hfill \\ \hfill A({e}_{2},{e}_{1})\hfill & \hfill A({e}_{2},{e}_{2})\hfill \end{array}\right)=r\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 0\hfill \end{array}\right)=rS,$

where $r=A({e}_{1},{e}_{2})$. The skew symmetric matrix $S$ has the property that its diagonal entries are all $0$. $S$ is called the $2\times 2$ *alternating* or *symplectic matrix*.

$A$ is called *non-singular* or *non-degenerate* if there exist a vectors $u,v\in V$ such that $A(u,v)\ne 0$. $u,v$ are necessarily non-zero. Note that the associated matrix $rS$ is non-singular iff $r\ne 0$ 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 $\mathcal{A}$.

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 $\mathcal{A}$. In other words, the associated matrix for $A$ has the block form

$\left(\begin{array}{cccc}\hfill S\hfill & \hfill \mathrm{\U0001d7ce}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill \mathrm{\U0001d7ce}\hfill \\ \hfill \mathrm{\U0001d7ce}\hfill & \hfill S\hfill & \hfill \mathrm{\cdots}\hfill & \hfill \mathrm{\U0001d7ce}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill \mathrm{\U0001d7ce}\hfill & \hfill \mathrm{\U0001d7ce}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill S\hfill \end{array}\right),\text{where}\mathrm{\U0001d7ce}=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$

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 |