It is called symmetric if for all . For variants on this, such as the property of being conjugate-symmetric, see inner product space.
Let be the real numbers. A symmetric bilinear form is called
positive definite if if .
negative definite if if .
A inner product on a real vector space is an example of a symmetric bilinear form. (For some authors, an inner product on a real vector space is precisely a positive definite symmetric bilinear form. Other authors relax the positive definiteness to nondegeneracy. Perhaps some authors even drop the nondegeneracy condition (citation?).)
If is of class , then the Hessian of at a point defines a symmetric bilinear form. It may be degenerate, but in Morse theory, a Morse function is a function such that the Hessian at each critical point is nondegenerate.