PlanetMath: non-degenerate quadratic form

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non-degenerate quadratic form

(Definition)

Let be a field of characteristic not 2. Then a quadratic form over a vector space (over a field ) is said to be non-degenerate, if its associated bilinear form:

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

is non-degenerate.

If we fix a basis $\boldsymbol{b}$ for , then can be written as

$\displaystyle Q(x)=x^TAx$

for some symmetric matrix over . Then it's not hard to see that is non-degenerate iff 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.

"non-degenerate quadratic form" is owned by CWoo.

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Other names:

non degenerate quadratic form, non singular quadratic form

Also defines:

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

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Cross-references: non-singular, iff, symmetric matrix, basis, fix, non-degenerate, bilinear form, vector space, quadratic form, characteristic, field
There are 4 references to this entry.

This is version 3 of non-degenerate quadratic form, born on 2005-02-25, modified 2006-02-26.
Object id is 6828, canonical name is NonDegenerateQuadraticForm.
Accessed 6684 times total.

Classification:

AMS MSC:	11E39 (Number theory :: Forms and linear algebraic groups :: Bilinear and Hermitian forms)
	15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
	47A07 (Operator theory :: General theory of linear operators :: Forms )

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