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 $𝒃$ 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