# non-degenerate quadratic form

Let $k$ be a field of characteristic not 2. Then a quadratic form^{} $Q$ over a vector space^{} $V$ (over a field $k$) is said to be , if its associated bilinear form^{}:

$$B(x,y)=\frac{1}{2}(Q(x+y)-Q(x)-Q(y))$$ |

is non-degenerate.

If we fix a basis $\bm{b}$ for $V$, then $Q(x)$ can be written as

$$Q(x)={x}^{T}Ax$$ |

for some symmetric matrix^{} $A$ over $k$. Then it’s not hard to see that $Q$ is non-degenerate iff $A$ is non-singular. Because of this, a non-degenerate quadratic form is also known as a *non-singular* quadratic form. A third name for a non-degenerate quadratic form is that of a *regular quadratic form*.

Title | non-degenerate quadratic form |

Canonical name | NondegenerateQuadraticForm |

Date of creation | 2013-03-22 15:05:58 |

Last modified on | 2013-03-22 15:05:58 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 6 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 15A63 |

Classification | msc 11E39 |

Classification | msc 47A07 |

Synonym | non degenerate quadratic form |

Synonym | non singular quadratic form |

Defines | non-degenerate quadratic form |

Defines | non-singular quadratic form |

Defines | regular quadratic form |