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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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