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Degenerate formFrom Wikipedia, the free encyclopedia

In mathematics, specifically linear algebra, a degeneratebilinear form (x,y) on a vector space Vis one such

that the map from to (the dual space of ) given by is not an isomorphism. An

equivalent definition when Vis finite-dimensional is that it has a non-trivial kernel: there exist some non-zero x

in Vsuch that

for all

A nondegenerate ornonsingular form is one that is not degenerate, meaning that is an

isomorphism, or equivalently in finite dimensions, if and only if

for all implies that x = 0.

IfVis finite-dimensional then, relative to some basis forV, a bilinear form is degenerate if and only if the

determinant of the associated matrix is zero if and only if the matrix is singular, and accordingly degenerate

forms are also called singular forms. Likewise, a nondegenerate form is one for which the associated matrix is

non-singular, and accordingly nondegenerate forms are also referred to as non-singular forms. These

statements are independent of the chosen basis.

There is the closely related notion of a unimodular form and a perfect pairing; these agree over fields but not

over general rings.

The most important examples of nondegenerate forms are inner products and symplectic forms. Symmetric

nondegenerate forms are important generalizations of inner products, in that often all that is required is that the

map be an isomorphism, not positivity. For example, a manifold with an inner product structure on

its tangent spaces is a Riemannian manifold, while relaxing this to a symmetric nondegenerate form yields a

pseudo-Riemannian manifold.

Infinite dimensions

Note that in an infinite dimensional space, we can have a bilinear form for which is

injective but not surjective. For example, on the space of continuous functions on a closed bounded interval, the

form

is not surjective: for instance, the Dirac delta functional is in the dual space but not of the required form. On the

other hand, this bilinear form satisfies

for all implies that

Terminology

If vanishes identically on all vectors it is said to be totally degenerate. Given any bilinear form on Vthe set

of vectors

nerate form - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Degene

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forms a totally degenerate subspace ofV. The map is nondegenerate if and only if this subspace is trivial.

Sometimes the words anisotropic, isotropic and totally isotropic are used for nondegenerate, degenerate and

totally degenerate respectively, although definitions of these latter words can vary slightly between authors.

Beware that a vector such that is called isotropic for the quadratic form associated with

the bilinear form and the existence of isotropic lines does not imply that the form is degenerate.

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nerate form - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Degene

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