nLab
bilinear form

Definitions

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

It is called symmetric if $⟨x,y⟩=⟨y,x⟩$ 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 $V\to V^{*}=\mathrm{hom}(V,k)$ is injective (a monomorphism).

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

• positive definite if $⟨x,x⟩>0$ if $x\ne 0$.

• negative definite if $⟨x,x⟩<0$ if $x\ne 0$.

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 $f:{ℝ}^n\to ℝ$ 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.

Related concepts

• quadratic form

• symplectic form