# Degenerate bilinear form

In mathematics, specifically linear algebra, a **degenerate bilinear form** *f*(*x*, *y*) on a vector space *V* is a bilinear form such that the map from *V* to *V*^{∗} (the dual space of *V*) given by *v* ↦ (*x* ↦ *f*(*x*, *v*)) is not an isomorphism. An equivalent definition when *V* is finite-dimensional is that it has a non-trivial kernel: there exist some non-zero *x* in *V* such that

If *V* is finite-dimensional then, relative to some basis for *V*, 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.

If for a quadratic form *Q* there is a vector *v* ∈ *V* such that *Q*(*v*) = 0, then *Q* is an isotropic quadratic form. If *Q* has the same sign for all vectors, it is a definite quadratic form or an **anisotropic quadratic form**.

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

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

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

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

forms a totally degenerate subspace of *V*. The map ƒ is nondegenerate if and only if this subspace is trivial.

Geometrically, an isotropic line of the quadratic form corresponds to a point of the associated quadric hypersurface in projective space. Such a line is additionally isotropic for the bilinear form if and only if the corresponding point is a singularity. Hence, over an algebraically 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.