Inmathematics, specificallylinear algebra, adegenerate bilinear formf (x,y ) on avector spaceV is abilinear form such that the map fromV toV^∗ (thedual space ofV ) given byv ↦ (x ↦f (x, v )) is not anisomorphism. An equivalent definition whenV isfinite-dimensional is that it has a non-trivial kernel: there exist some non-zerox inV such that

${\displaystyle f(x,y)=0}$ for all${\displaystyle y\in V.}$

Anondegenerate ornonsingular form is abilinear form that is not degenerate, meaning that${\displaystyle v\mapsto (x\mapsto f(x,v))}$ is anisomorphism, or equivalently in finite dimensions,if and only if^[1]

${\displaystyle f(x,y)=0}$ for all${\displaystyle y\in V}$ implies that${\displaystyle x=0}$.

The most important examples of nondegenerate forms areinner products andsymplectic forms.Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map${\displaystyle V\to V^{\ast }}$ be an isomorphism, not positivity. For example, amanifold with an inner product structure on itstangent spaces is aRiemannian manifold, while relaxing this to a symmetric nondegenerate form yields apseudo-Riemannian manifold.

IfV is finite-dimensional then, relative to somebasis forV, a bilinear form is degenerate if and only if thedeterminant of the associatedmatrix is zero – if and only if the matrix issingular, and accordingly degenerate forms are also calledsingular forms. Likewise, a nondegenerate form is one for which the associated matrix isnon-singular, and accordingly nondegenerate forms are also referred to asnon-singular forms. These statements are independent of the chosen basis.

If for aquadratic formQ there is a non-zero vectorv ∈V such thatQ(v) = 0, thenQ is anisotropic quadratic form. IfQ has the same sign for all non-zero vectors, it is adefinite quadratic form or ananisotropic quadratic form.

There is the closely related notion of aunimodular form and aperfect pairing; these agree overfields but not over generalrings.

The study of real, quadratic algebras shows the distinction between types of quadratic forms. The productzz* is a quadratic form for each of thecomplex numbers,split-complex numbers, anddual numbers. Forz =x + εy, the dual number form isx² which is adegenerate quadratic form. The split-complex case is an isotropic form, and the complex case is a definite form.

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${\displaystyle V\to V^{\ast }}$ 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.

Note that in an infinite-dimensional space, we can have a bilinear form ƒ for which${\displaystyle v\mapsto (x\mapsto f(x,v))}$ isinjective but notsurjective. For example, on the space ofcontinuous functions on a closed boundedinterval, the form

${\displaystyle f(\varphi ,\psi )=\int \psi (x)\varphi (x)dx}$

is not surjective: for instance, theDirac delta functional is in the dual space but not of the required form. On the other hand, this bilinear form satisfies

${\displaystyle f(\varphi ,\psi )=0}$ for all${\displaystyle \varphi }$ implies that${\displaystyle \psi =0.}$

In such a case where ƒ satisfies injectivity (but not necessarily surjectivity), ƒ is said to beweakly nondegenerate.

Iff vanishes identically on all vectors it is said to be totally degenerate. Given any bilinear formf onV the set of vectors

${\displaystyle \{x\in V\mid f(x,y)=0\text{ for all }y\in V\}}$

forms a totally degeneratesubspace ofV. The mapf is nondegenerate if and only if this subspace is trivial.

Geometrically, anisotropic line of the quadratic form corresponds to a point of the associatedquadric hypersurface inprojective space. Such a line is additionally isotropic for the bilinear form if and only if the corresponding point is asingularity. Hence, over analgebraically closed field,Hilbert's Nullstellensatz guarantees that the quadratic form always has isotropic lines, while the bilinear form has them if and only if the surface is singular.

• Indefinite inner product space – generalization of Hilbert space with indefinite signaturePages displaying wikidata descriptions as a fallback
• Dual system
• Linear form – Linear map from a vector space to its field of scalars

• Bilinear forms
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