A bilinear form is simply a linear map$\langle -,-\rangle\colon V \otimes V \to k$ out of a tensor product of $k$-modules into the ring$k$ (typically taken to be a field).

It is called symmetric if $\langle x,y\rangle = \langle y,x\rangle$ for all $x,y \in V$. For variants on this, such as the property of being conjugate-symmetric, see inner product space.

It is called nondegenerate if the mate$V \to V^\ast = \hom(V, k)$ is injective (a monomorphism).

Let $k = \mathbb{R}$ be the real numbers. A symmetric bilinear form is called

positive definite if $\langle x,x\rangle \gt 0$ if $x \neq 0$.

negative definite if $\langle x,x\rangle \lt 0$ if $x \neq 0$.

Examples

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 $f \colon \mathbb{R}^n \to \mathbb{R}$ is of class $C^2$, then the Hessian of $f$ at a point defines a symmetric bilinear form. It may be degenerate, but in Morse theory, a Morse function is a $C^2$ function such that the Hessian at each critical point is nondegenerate.