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# 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 *non-degenerate*, if its associated bilinear form:

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

is non-degenerate.

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

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*.

Defines:

non-degenerate quadratic form, non-singular quadratic form, regular quadratic form

Synonym:

non degenerate quadratic form, non singular quadratic form

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Definition

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## Mathematics Subject Classification

15A63*no label found*11E39

*no label found*47A07

*no label found*

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