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单词 NondegenerateQuadraticForm
释义

non-degenerate quadratic form


Let k be a field of characteristic not 2. Then a quadratic formMathworldPlanetmath Q over a vector spaceMathworldPlanetmath V (over a field k) is said to be , if its associated bilinear formPlanetmathPlanetmath:

B(x,y)=12(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)=xTAx

for some symmetric matrixMathworldPlanetmath 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.

Titlenon-degenerate quadratic form
Canonical nameNondegenerateQuadraticForm
Date of creation2013-03-22 15:05:58
Last modified on2013-03-22 15:05:58
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id6
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 15A63
Classificationmsc 11E39
Classificationmsc 47A07
Synonymnon degenerate quadratic form
Synonymnon singular quadratic form
Definesnon-degenerate quadratic form
Definesnon-singular quadratic form
Definesregular quadratic form
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