Nondegenerate form

In linear algebra, a nondegenerate form or nonsingular form is a bilinear form that is not degenerate, meaning that ${\displaystyle v\mapsto (x\mapsto f(x,v))}$ is an isomorphism, or equivalently in finite dimensions, if and only if

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

Examples

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

See also

• Degenerate bilinear form