Proportionality: Difference between revisions

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-{{Unreferenced|date=August 2008}}
-{{Other uses|Degeneracy (disambiguation){{!}}Degeneracy}}
-In [[mathematics]], specifically [[linear algebra]], a '''degenerate''' [[bilinear form]] ƒ(''x,y'') on a [[vector space]] ''V'' is one such that the map from <math>V</math> to <math>V^*</math> (the [[dual space]] of <math>V</math>) given by <math>v \mapsto (x \mapsto f(x,v))</math> is not an [[isomorphism]].  An equivalent definition when ''V'' is finite-dimensional is that it has a non-trivial [[kernel (algebra)|kernel]]: there exist some non-zero ''x'' in ''V'' such that
-:<math>f(x,y)=0\,</math> for all <math>y \in V.</math>
-A '''nondegenerate''' or '''nonsingular''' form is one that is not degenerate, meaning that <math>v \mapsto (x \mapsto f(x,v))</math>  is an [[isomorphism]], or equivalently in finite dimensions, if and only if
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-:<math>f(x,y)=0\,</math> for all <math>y \in V</math> implies that ''x'' = 0.
-If ''V'' is [[finite-dimensional]] then, relative to some [[basis (linear algebra)|basis]] for ''V'', a bilinear form is degenerate if and only if the [[determinant]] of the associated [[matrix (mathematics)|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 matrix|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 product]]s and [[symplectic form]]s. Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map <math>V \to V^*</math> 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 ƒ for which <math>v \mapsto (x \mapsto f(x,v))</math> is [[injective]] but not [[surjective]].  For example, on the space of [[continuous function]]s on a closed bounded interval, the form
-:<math> f(\phi,\psi) = \int\psi(x)\phi(x) dx</math>
-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
-:<math>f(\phi,\psi)=0\,</math> for all <math>\,\phi</math> implies that <math>\psi=0.\,</math>
-==Terminology==
-If ƒ vanishes identically on all vectors it is said to be ''' totally degenerate'''. Given any bilinear form ƒ on ''V'' the set of vectors
-:<math>\{x\in V \mid f(x,y) = 0 \mbox{ for all } y \in V\}</math>
-forms a totally degenerate [[linear subspace|subspace]] of ''V''. The map ƒ 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 <math>x\in V</math> such that  <math>f(x,x)=0</math> is called isotropic for the [[isotropic quadratic form|quadratic form]] associated with the bilinear form <math>f</math> and the existence of isotropic lines does not imply that the form is degenerate.
-[[Category:Bilinear forms]]
-[[pl:Forma dwuliniowa#Własności]]

Proportionality: Difference between revisions

Latest revision as of 04:25, 18 April 2014

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